Deformation Quantization for Actions of Kählerian Lie Groups
aa r X i v : . [ m a t h . OA ] J a n Deformation Quantization for Actions ofK¨ahlerian Lie Groups
Pierre BieliavskyVictor Gayral
University of Louvain, Belgium
E-mail address : [email protected] University of Reims, France
E-mail address : [email protected] Mathematics Subject Classification.
Primary 22E30, 46L87, 81R60, 58B34,81R30, 53C35, 32M15, 53D55
Key words and phrases.
Strict deformation quantization, Symmetric spaces,Representation theory of Lie groups, Deformation of C ∗ -algebras, Symplectic Liegroups, Coherent states, Noncommutative harmonic analysis, NoncommutativegeometryWork supported by the Belgian Interuniversity Attraction Pole (IAP) within theframework “Dynamics, Geometry and Statistical Physics” (DYGEST). Abstract.
Let B be a Lie group admitting a left-invariant negatively curvedK¨ahlerian structure. Consider a strongly continuous action α of B on a Fr´echetalgebra A . Denote by A ∞ the associated Fr´echet algebra of smooth vectors forthis action. In the Abelian case B = R n and α isometric, Marc Rieffel provedin [ ] that Weyl’s operator symbol composition formula (the so called Moyalproduct) yields a deformation through Fr´echet algebra structures { ⋆ αθ } θ ∈ R on A ∞ . When A is a C ∗ -algebra, every deformed Fr´echet algebra ( A ∞ , ⋆ αθ ) ad-mits a compatible pre- C ∗ -structure, hence yielding a deformation theory atthe level of C ∗ -algebras too. In this memoir, we prove both analogous state-ments for general negatively curved K¨ahlerian groups. The construction relieson the one hand on combining a non-Abelian version of oscillatory integral ontempered Lie groups with geometrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand,in establishing a non-Abelian version of the Calder´on-Vaillancourt Theorem.In particular, we give an oscillating kernel formula for WKB-star products onsymplectic symmetric spaces that fiber over an exponential Lie group. ontents Introduction 1Notations and conventions 7Chapter 1. Oscillatory integrals 91.1. Symbol spaces 101.2. Tempered pairs 211.3. An oscillatory integral for admissible tempered pairs 291.4. A Fubini Theorem for semi-direct products 321.5. A Schwartz space for tempered pairs 341.6. Bilinear mappings from the oscillatory integral 36Chapter 2. Tempered pairs for K¨ahlerian Lie groups 412.1. Pyatetskii-Shapiro’s theory 412.2. Geometric structures on elementary normal j -groups 432.3. Tempered pair for elementary normal j -groups 462.4. Tempered pairs for normal j -groups 51Chapter 3. Non-formal star-products 573.1. Star-products on normal j -groups 573.2. An oscillatory integral formula for the star-product 60Chapter 4. Deformation of Fr´echet algebras 654.1. The deformed product 654.2. Relation with the fixed point algebra 714.3. Functorial properties of the deformed product 72Chapter 5. Quantization of polarized symplectic symmetric spaces 775.1. Polarized symplectic symmetric spaces 795.2. Unitary representations of symmetric spaces 845.3. Locality and the one-point phase 885.4. Unitarity and midpoints for elementary spaces 895.5. The ⋆ -product as the composition law of symbols 945.6. The three-point kernel 965.7. Extensions of polarization quadruples 99Chapter 6. Quantization of K¨ahlerian Lie groups 1036.1. The transvection quadruple of an elementary normal j -group 1036.2. Quantization of elementary normal j -groups 1086.3. Quantization of normal j -groups 112Chapter 7. Deformation of C ∗ -algebras 117 iiiv CONTENTS C ∗ -norm 1347.6. Functorial properties of the deformation 1397.7. Invariance of the K -theory 140Bibliography 151 ntroduction The general idea of deforming a given theory by use of its symmetries goes backto Drinfel’d. One paradigm being that the data of a
Drinfel’d twist based on a bi-algebra acting on an associative algebra A , produces an associative deformationof A . In the context of Lie theory, one considers for instance the category ofmodule-algebras over the universal enveloping algebra U ( g ) of the Lie algebra g ofa given Lie group G . In that situation, the notion of Drinfel’d twist is in a oneto one correspondence with the one of left-invariant formal star-product ⋆ ν on thespace of formal power series C ∞ ( G )[[ ν ]], see [ ]. Disposing of such a twist, every U ( g )-module-algebra A may then be formally deformed into an associative algebra A [[ ν ]]. It is important to observe that, within this situation, the symplectic leaf B through the unit element e of G in the characteristic foliation of the (left-invariant)Poisson structure directing the star-product ⋆ ν , always consists of an immersedLie subgroup of G . The Lie group B therefore carries a left-invariant symplectic structure. This stresses the importance of symplectic Lie groups (i.e . connected Liegroups endowed with invariant symplectic forms) as semi-classical approximationsof Drinfel’d twists attached to Lie algebras.In the present memoir, we address the question of designing non-formal Drin-fel’d twists for actions of symplectic Lie groups B that underly negatively curved K¨ahlerian Lie groups , i.e . Lie groups that admit a left-invariant K¨ahlerian structureof negative curvature. These groups exactly correspond to the normal j -algebrasdefined by Pyatetskii-Shapiro in his work on automorphic forms [ ]. In particular,this class of groups contains all Iwasawa factors AN of Hermitian type simple Liegroups G = KAN .Roughly speaking, one looks for a smooth one-parameter family of complexvalued smooth two-point functions on the group, { K θ } θ ∈ R ⊂ C ∞ ( B × B , C ), withthe property that, for every strongly continuous and isometric action α of B on a C ∗ -algebra A , the following formula(0.1) a ⋆ αθ b := Z B × B K θ ( x, y ) α x ( a ) α y ( b ) d x d y , defines a one-parameter deformation of the C ∗ -algebra A .The above program was realized by Marc Rieffel in the particular case of theAbelian Lie group B = R n in [ ]. More precisely, Rieffel proved that for any strongly continuous and isometric action of R n on any Fr´echet algebra A , theassociated Fr´echet subalgebra A ∞ of smooth vectors for this action, is deformedby the rule (0.1), where the two-point kernel there, consists of the Weyl symbolcomposition kernel: K θ ( x, y ) := θ − n exp (cid:8) iθ ω ( x, y ) (cid:9) , associated to a translation invariant symplectic structure ω on R n . The associatedstar-product therefore corresponds here to Moyal’s product. In the special casewhere the Fr´echet algebra A is a C ∗ -algebra, Rieffel also constructed a deformed C ∗ -structure, so that ( A ∞ , ⋆ αθ ) becomes a pre- C ∗ -algebra, which in turn yieldsa deformation theory at the level of C ∗ -algebras too. Many further results havebeen proved then (for example continuity of the field of deformed C ∗ -algebras [ ],invariance of the K -theory [ ]...), and many applications have been found (forinstance in locally compact quantum groups [ ], quantum fields theory [ , ],spectral triples [ ]...).In this memoir, we investigate the deformation theory of C ∗ -algebras endowedwith an isometric action of a negatively curved K¨ahlerian Lie group. Most of theresults we present here are of a pure analytical nature. Indeed, once a family { K θ } θ ∈ R of associative (i.e . such that the associated deformed product (0.1) is atleast formally associative) two-point functions has been found, in order to givea precise meaning to the associated multiplication rule, there is no doubt thatthe integrals in (0.1) need to be interpreted in a suitable (here oscillatory) sense.Indeed, there is no reason to expect the two-point function K θ to be integrable: itis typically not even bounded in the non-Abelian case! Thus, already in the case ofan isometric action on a C ∗ -algebra, we have to face a serious analytical difficulty.We stress that contrarily to the case of R n , in the situation of a non-Abelian groupaction, this is an highly non-trivial feature of our deformation theory.The memoir is organized as follows.In chapter 1, we start by introducing non-Abelian and unbounded versions ofFr´echet-valued symbol spaces on a Lie group G , with Lie algebra g : B µ ( G, E ) := n f ∈ C ∞ ( G, E ) : ∀ X ∈ U ( g ) , ∀ j ∈ N , ∃ C > k e Xf k j ≤ C µ j o , where E is a Fr´echet space, µ := { µ j } j ∈ N is a countable family of specific posi-tive functions on G , called weights (see Definition 1.1) affiliated to a countable setof semi-norms {k . k j } j ∈ N defining the Fr´echet topology on E and where e X is theleft-invariant differential operator on G associated to an element X ∈ U ( g ). Forexample, B ( G, C ) consists of the smooth vectors for the right regular representa-tion of G on the space of bounded right-uniformly continuous functions on G (theuniform structures on G are generally not balanced in our non-Abelian situation)and it coincides with Laurent Schwartz’s space B when G = R n . We shall alsomention that function spaces on Lie groups of a similar type are considered in [ ]and in [ ] in the context of actions of R d on locally convex algebras.We then define a notion of oscillatory integrals on Lie groups G that are endowedwith a specific type of smooth function S ∈ C ∞ ( G, R ) (see Definitions 1.17, 1.22and 1.24). We call such a pair ( G, S ) an admissible tempered pair . The main resultof this chapter is that associated to an admissible tempered pair (
G, S ), and givena growth-controlled function m , the oscillatory integral D ( G, E ) → E , F Z G m e iS F , canonically extends from D ( G, E ), the space of smooth compactly supported func-tions, to our symbol space B µ ( G, E ). This construction is explained in Definition NTRODUCTION 3 normal j -group B (i.e . a connected sim-ply connected Lie group whose Lie algebra is a normal j -algebra—see Definition2.1). The main result of this chapter, Theorem 2.35, shows that its square B × B canonically underlies an admissible tempered pair ( B × B , S B can ). When elementary,every normal j -group has a canonical simply transitive action on a specific solv-able symplectic symmetric space. The two-point function S B can we consider herecomes from an earlier work of one of us. It consists of the sum of the phases S S j can of the oscillatory kernels associated to invariant star-products on solvable sym-plectic symmetric space [ , ], in the Pyatetskii-Shapiro decomposition [ ] of anormal j -group B into a sequence of split extensions of elementary normal j -factors: B = ( . . . (( S ⋉ S ) ⋉ S ) ⋉ . . . ) ⋉ S N . The two-point phase function S S can in theelementary case, then consists of the symplectic area of the unique geodesic trianglein S (viewed as a solvable symplectic symmetric space), whose geodesic edges admit e, x and y as midpoints ( e denotes the unit element of the group S ): S S can ( x , x ) := Area (cid:0) Φ − S ( e, x , x ) (cid:1) , with Φ S : S → S , ( x , x , x ) (cid:0) mid ( x , x ) , mid ( x , x ) , mid ( x , x ) (cid:1) , where mid ( x, y ) denotes the geodesic midpoint between x and y in S (again uniquelydefined in our situation).In chapter 3, we consider an arbitrary normal j -group B , and define the above-mentioned oscillatory kernels K θ simply by tensorizing oscillating kernels foundin [ ] on elementary j -factors. The resulting kernel has the form K θ = θ − dim B m B can exp (cid:8) iθ S B can (cid:9) , where S B can is the two-point phase mentioned in the description of chapter 2 above,and m B can = m S can ⊗ · · ·⊗ m S N can , where m S j can = Jac / − S j denotes the square root of theJacobian of the “medial triangle” map Φ − S . In particular, it defines an oscillatoryintegral on every symbol space of the type B µ ( B × B , B ν ( B , E )). When valued ina Fr´echet algebra A , this yields a non-perturbative and associative star-product ⋆ θ on the union of all symbol spaces B µ ( B , A ).In chapter 4, we consider any tempered action of a normal j -group B on aFr´echet algebra A . By tempered action we mean a strongly continuous action α of B by automorphisms on A , such that for every semi-norm k . k j there is aweight (“tempered” for a suitable notion of temperedness) µ αj such that k α g ( a ) k j ≤ µ αj ( g ) k a k j for all a ∈ A and g ∈ B . In that case, the space of smooth vectors A ∞ for α naturally identifies with a subspace of B ˆ µ ( B , A ∞ ) (where ˆ µ is affiliated to µ α = { µ αj } j ∈ N ) through the injective map: α : A ∞ → B ˆ µ ( B , A ∞ ) , a [ g α g ( a )] . We stress that even in the case of an isometric action, and contrarily to the Abeliansituation, the map α always takes values in a symbol space B µ , for a non-trivial INTRODUCTION sequence of weights µ . This explains why the non-Abelian framework forces toconsider such symbol spaces. Applying the results of chapter 3 to this situation,we get a new associative product on A ∞ defined by the formula a ⋆ αθ b := (cid:0) α ( a ) ⋆ θ α ( b ) (cid:1) ( e ) . Then, the main result of this chapter, stated as Theorem 4.8 in the text, is thefollowing fact:
Universal Deformation Formula for Actions of K¨ahlerian Lie Groups onFr´echet Algebras:
Let ( A , α, B ) be a Fr´echet algebra endowed with a tempered action of a normal j -group. Then, ( A ∞ , ⋆ αθ ) is an associative Fr´echet algebra, (abusively) called theFr´echet deformation of A . Following the terminology introduced in [ ], the word “universal” refers to thefact that our deformation procedure applies to any Fr´echet algebra the K¨ahlerianLie group acts on. The word “universal” does not refer to the possibility that ourconstruction might yield all such deformation procedures valid for a given K¨ahlerianLie group. However, regarding the last sentence, the following remark can never-theless be made. In order to get rid of technicalities let us consider, for a shortmoment, the purely formal framework of formal Drinfel’d twists based on the bi-algebra underlying the enveloping algebra U ( b ) of the Lie group B . Given such aDrinfel’d twist F ∈ U ( b ) ⊗ U ( b )[[ ν ]], one defines the internal symmetry of the twistas the group G ( F ), consisting of diffeomorphisms that preserve the twist: G ( F ) := { ϕ : B → B | ϕ ⋆ ˜ F = ˜ F } , where ˜ F denotes the left-invariant (formal sequence of) operator(s) associated tothe twist F . Within the present work we treat (all) the situations where, in theelementary normal case, the internal symmetry equals the automorphism group ofa symmetric symplectic space structure on B whose underlying affine connectionconsists in the canonical torsion free invariant connection on the group B (see [ ]).The rest of the memoir is devoted to the construction of a pre- C ∗ -structure on( A ∞ , ⋆ αθ ), in the case A is a C ∗ -algebra. The method we use is a generalization ofUnterberger’s Fuchs calculus [ ] and fits within the general framework of “Moyalquantizer” as defined in [ ] (see also [ , Section 3.5]).In chapter 5, we define a special class of symplectic symmetric spaces whichnaturally give rise to explicit WKB-quantizations (i.e . invariant star-products rep-resentable through oscillatory kernels) that underlie pseudo-differential operatorcalculus. Roughly speaking, an elementary symplectic symmetric space is a sym-plectic symmetric space that consists of the total space of a fibration in flat fibersover a Lie group Q of exponential type. In that case, a variant of Kirillov’s orbitmethod yields a unitary and self-adjoint representation on an Hilbert-space H , ofthe symmetric space M : Ω : M → U sa ( H ) , NTRODUCTION 5 with associated “quantization rule”:Ω : L ( M ) → B ( H ) , F Ω( F ) := Z M F ( x )Ω( x ) d x . Weighting the above mapping by the multiplication by a (growth controlled) func-tion m defined on the base Q yields a pair of adjoint maps:Ω m : L ( M ) → L ( H ) and σ m : L ( H ) → L ( M ) , where L ( H ) denotes the Hilbert space of Hilbert-Schmidt operators on H . Bothof the above maps are equivariant under the whole automorphism group of M .Note that this last feature very much contrasts with the usual notion of coherent-state quantization for groups (as opposed to symmetric spaces). The corresponding“Berezin transform” B m := σ m ◦ Ω m is explicitly controlled. In particular, wheninvertible, the associated star-product F ⋆ F := B − m σ m (Ω m ( F )Ω m ( F )) is of os-cillatory (WKB) type and its associated kernel is explicitly determined. Note that,because entirely explicit, this chapter yields a proof of Weinstein’s conjectural formfor star-product WKB-kernels on symmetric spaces [ ] in the situation consid-ered here. The chapter ends with considerations on extending the construction tosemi-direct products.Chapter 6 is entirely devoted to applying the construction of chapter 5 tothe particular case of K¨ahlerian Lie groups with negative curvature. Such a Liegroup is always a normal j -group is the sense of Pyatetskii-Shapiro. Each of itselementary factors admits the structure of an elementary symplectic symmetricspace. Accordingly to [ ], the obtained non-formal star products coincide with theone described in chapters 2-4.Chapter 7 deals with the deformation theory for C ∗ -algebras. We eventuallyprove the following statement: Universal Deformation Formula for Actions of K¨ahlerian Lie Groups on C ∗ -Algebras: Let ( A, α, B ) be a C ∗ -algebra endowed with a strongly continuous and isometricaction of a normal j -group. Then, there exists a canonical C ∗ -norm on the Fr´echetalgebra ( A ∞ , ⋆ αθ ) . Its C ∗ -closure is (abusively) called the C ∗ -deformation of A . The above statement follows from a non-Abelian generalization of the Calder´on-Vaillancourt Theorem in the context of the usual Weyl pseudo-differential calculuson R n (see Theorem 7.20). Our result asserts that the element Ω m ( F ) associatedto a function F in B ( B , A ) naturally consists of an element of the spatial tensorproduct of A by B ( H ). Its proof relies on a combination of a resolution of theidentity obtained from wavelet analysis considerations (see section 7.1) and furtherproperties of our oscillatory integral defined in chapter 1. We finally prove that the K -theory is an invariant of the deformation. Acknowledgments
We warmly thank the referee for his positive criticism.His comments, remarks and suggestions have greatly improved this work. otations and conventions
Given a Lie group G , with Lie algebra g , we denote by d G ( g ) a left invariantHaar measure. In the non-unimodular case, we consider the modular function ∆ G ,defined by the relation: d G ( g )∆ G ( g ) := d G ( g − ) . Unless otherwise specified, L p ( G ), p ∈ [1 , ∞ ], will always denote the Lebesgue spaceassociated with the choice of a left-invariant Haar measure made above. We alsodenote by D ( G ) the space of smooth compactly supported functions on G and by D ′ ( G ) the dual space of distributions.We use the notations L ⋆ and R ⋆ , for the left and right regular actions: L ⋆g f ( g ′ ) := f ( g − g ′ ) , R ⋆g f ( g ′ ) := f ( g ′ g ) . By e X and X , we mean the left-invariant and right-invariant vector fields on G associated to the elements X and − X of g : e X := ddt (cid:12)(cid:12)(cid:12) t =0 R ⋆e tX , X := ddt (cid:12)(cid:12)(cid:12) t =0 L ⋆e tX . Given a element X of the universal enveloping algebra U ( g ) of g , we adopt the samenotations e X and X for the associated left- and right-invariant differential operatorson G . More generally, if α is an action of G on a topological vector space E , weconsider the infinitesimal form of the action, given for X ∈ g by: X α ( a ) := ddt (cid:12)(cid:12)(cid:12) t =0 α e tX ( a ) , a ∈ E ∞ , and extended to the whole universal enveloping algebra U ( g ), by declaring that themap U ( b ) → End ( A ∞ ), X X α is an algebra homomorphism. Here, E ∞ denotesthe set of smooth vectors for the action: E ∞ := (cid:8) a ∈ E : [ g α g ( a )] ∈ C ∞ ( G, E ) (cid:9) . Let ∆ U be the ordinary co-product of U ( g ). We make use of the Sweedlernotation: ∆ U ( X ) = X ( X ) X (1) ⊗ X (2) ∈ U ( g ) ⊗ U ( g ) , X ∈ U ( g ) , and accordingly, for f , f ∈ C ∞ ( G ) and X ∈ U ( g ), we write(0.2) e X ( f f ) = X ( X ) (cid:0) e X (1) f (cid:1) (cid:0) e X (2) f (cid:1) , X ( f f ) = X ( X ) (cid:0) X (1) f (cid:1) (cid:0) X (2) f (cid:1) . More generally, we use the notation(∆ U ⊗ Id ) ◦ ∆ U ( X ) = X ( X ) X ( X (1) ) ( X (1) ) (1) ⊗ ( X (1) ) (2) ⊗ X (2) (0.3) =: X ( X ) X (11) ⊗ X (12) ⊗ X (2) , and obvious generalization of it.To a fixed ordered basis { X , . . . , X m } of the Lie algebra g , we associate a PBWbasis of U ( g ):(0.4) { X β , β ∈ N m } , X β := X β X β . . . X β m m . This induces a filtration U ( g ) = [ k ∈ N U k ( g ) , U k ( g ) ⊂ U l ( g ) , k ≤ l , in terms of the subsets(0.5) U k ( g ) := n X | β |≤ k C β X β , C β ∈ R o , k ∈ N , where | β | := β + · · · + β m . For β, β , β ∈ N m , we define the ‘structure constants’ ω β ,β β ∈ R of U ( g ), by(0.6) X β X β = X | β |≤| β | + | β | ω β ,β β X β ∈ U | β | + | β | ( g ) . We endow the finite dimensional vector space U k ( g ), with the ℓ -norm | . | k withinthe basis { X β , | β | ≤ k } :(0.7) | X | k := X | β |≤ k | C β | if X = X | β |≤ k C β X β ∈ U k ( g ) . We observe that the family of norms {| . | k } k ∈ N is compatible with the filtered struc-ture of U ( g ), in the sense that if X ∈ U k ( g ), then | X | k = | X | l whenever l ≥ k .Considering a subspace V ⊂ g , we also denote by U ( V ) the unital subalgebra of U ( g ) generated by V : U ( V ) = span n X X . . . X n : X j ∈ V , n ∈ N o , (0.8)that we may filtrate using the induced filtration of U ( g ). We also observe thatthe co-product preserves the latter subalgebras, in the sense that ∆ U (cid:0) U ( V ) (cid:1) ⊂U ( V ) ⊗ U ( V ).Regarding the uniform structures on a locally compact group G , we say that afunction f : G → C is right (respectively left) uniformly continuous if for all ε > U , an open neighborhood of the neutral element e , such that for all( g, h ) ∈ G × G we have | f ( g ) − f ( h ) | ≤ ε , whenever g − h ∈ U (cid:0) respectively hg − ∈ U (cid:1) . We call a Lie group G (with Lie algebra g ) exponential, if the exponential mapexp : g → G is a global diffeomorphism.Let f , f be two real valued functions on G . We say that f and f have thesame behavior, that we write f ≍ f , when there exist 0 < c ≤ C such that for all x ∈ G , we have cf ( x ) ≤ f ( x ) ≤ Cf ( x ).HAPTER 1 Oscillatory integrals
This chapter is the most technical part of this memoir. To outline its content,we shall first explain the situation on which the general theory is designed. Consider B a simply connected Lie group endowed with a left invariant K¨ahlerian structurewith negative curvature. Let also α be a strongly continuous and isometric actionof B on a C ∗ -algebra A . In this context and from previous works of one of us, wehave at our disposal two functions m can ∈ C ∞ ( B × B , R ∗ + ) and S can ∈ C ∞ ( B × B , R )such that setting for θ ∈ R ∗ K θ = θ − dim B m can exp (cid:8) iθ S can (cid:9) ∈ C ∞ ( B × B , C ) , the following formula(1.1) a ⋆ αθ b := Z B × B K θ ( x, y ) α x ( a ) α y ( b ) d x d y , a, b ∈ A , formally defines a one-parameter family of associative algebra structures on A .Since the function m can is unbounded and since the map α ( a ) := [ x α x ( a )]is constant in norm, the only hope to go beyond the formal level is to define theintegral sign in (1.1) in an oscillatory way. What we are precisely looking for, is apair ( A , D ), where • A is a dense and α -stable Fr´echet subalgebra of A with a topology(finer than the uniform one) determined by a countable set of semi-norms {k . k j } j ∈ N • D := { D j } j ∈ N is a countable family of differential operators on B × B , suchthat for all a, b ∈ A and all j ∈ N , the image under D j of the map [( x, y ) ∈ B × B m can ( x, y ) α x ( a ) α y ( b ) ∈ A ] belongs to L ( B × B , ( A , k . k j )) and,denoting by D ∗ j the formal adjoint of D j , such that D ∗ j exp (cid:8) iθ S can (cid:9) = exp (cid:8) iθ S can (cid:9) . Once such a pair is found, there is a clear way to give a meaning of ⋆ αθ on A ,namely by the k . k j -absolutely convergent A -valued integral: a ⋆ αθ b := θ − dim B Z B × B exp (cid:8) iθ S can ( x, y ) (cid:9) D j ; x,y (cid:0) m can ( x, y ) α x ( a ) α y ( b ) (cid:1) d x d y . There is an obvious candidate for A , which is A ∞ , the Fr´echet subalgebra of A consisting of smooth vectors for the action α : A ∞ := (cid:8) a ∈ A : α ( a ) = [ x α x ( a )] ∈ C ∞ ( B , A ) (cid:9) . Recall that by the strong continuity assumption of the action, A ∞ is dense in A .Then, the crucial observation, proved in Lemma 4.5, is that we have a continuousembedding: α ⊗ α : A ∞ × A ∞ → B µ ( B × B , A ∞ ) , ( a, b ) [( g, h ) α g ( a ) α h ( b )] ,
90 1. OSCILLATORY INTEGRALS where the symbol space B µ ( B × B , A ∞ ) is defined in (1.9) and where µ = { µ j } j ∈ N is a family of unbounded functions on B × B , called weights (see Definition 1.1below). We should already stress that it is because the group B is non Abelian(and noncompact) that we are forced to consider such symbol spaces associated tounbounded weights. Therefore, the problem becomes the construction of a familyof differential operators D = { D j } j ∈ N , such that for all j ∈ N , we have D j : B µ j ( B × B , A ∞ ) → L (cid:0) B × B , ( A ∞ , k . k j ) (cid:1) , and D ∗ j exp (cid:8) iθ S can (cid:9) = exp (cid:8) iθ S can (cid:9) . The construction of the family of differential operators D , and therefore of theassociated notion of oscillatory integral, is rather involved and this is the wholesubject of this chapter. Since in this memoir we eventually need an oscillatoryintegral for different groups (for B × B in chapters 3 and 4 and for B in chapter7) and since we intend to apply the whole ideas of the present construction tosituations involving other groups and other kernels, we have decided to formulatethe results of this chapter in the greatest possible generality. In this preliminary section, we consider a non-Abelian, weighted and Fr´echetspace valued version of the Laurent Schwartz space B of smooth functions that,together with all of their derivatives, are bounded. For reasons that will becomeclear later (cf . chapter 5), we refer to such function spaces as symbol spaces . Theyare constructed out of a family of specific functions on a Lie group G , that we call weights . The prototype of a weight for a non-Abelian Lie group is given in Definition1.3. The key properties of these symbol spaces are established in Lemmas 1.8 and1.12. In Lemma 1.15, we show on a fundamental example, how such spaces naturallyappear in the context of non-Abelian Lie group actions. Definition . Consider a connected real Lie group G with Lie algebra g . Anelement µ ∈ C ∞ ( G, R ∗ + ) is called a weight if it satisfies the following properties:(i) For every element X ∈ U ( g ), there exist C L , C R > | e Xµ | ≤ C L µ and | X µ | ≤ C R µ . (ii) There exist C, L, R ∈ R ∗ + such that for all g, h ∈ G we have: µ ( gh ) ≤ C µ ( g ) L µ ( h ) R . A pair (
L, R ) as in item (ii) is called a sub-multiplicative degree of the weight µ . A weight with sub-multiplicative degree (1 ,
1) is called a sub-multiplicativeweight . Remark . For µ ∈ C ∞ ( G ), we set µ ∨ ( g ) := µ ( g − ). Then, from the relation e X µ ∨ = ( X µ ) ∨ for all X ∈ U ( g ), we see that µ is a weight of sub-multiplicativedegree ( L, R ) if and only if µ ∨ is a weight of sub-multiplicative degree ( R, L ).Moreover, a product of two weights is a weight and a (positive) power of a weight isa weight. Also, the tensor product of two weights is a weight on the direct productgroup. .1. SYMBOL SPACES 11
In the following, we construct a canonical and non-trivial weight for non-Abelian Lie groups. This specific weight is an important object as it will naturallyand repeatedly appear in all our analysis (see for instance Lemmas 1.15, 1.42, 1.49,4.3 and 4.5).
Definition . Choosing an Euclidean norm | . | on g , for x ∈ G , we let | Ad x | be the operator norm of the adjoint action of G on g . The function d G : G → R ∗ + , x q | Ad x | + | Ad x − | , is called the modular weight of G . Lemma . The modular weight d G is a sub-multiplicative weight on G . Proof.
To prove that d G is a weight, we start from the relations for X ∈ g and x ∈ G : e X | Ad x | = 2 sup Y ∈ g , | Y | =1 h Ad x ◦ ad X ( Y ) , Ad x ( Y ) i , e X | Ad x − | = − Y ∈ g , | Y | =1 h ad X ◦ Ad x − ( Y ) , Ad x − ( Y ) i ,X | Ad x | = − Y ∈ g , | Y | =1 h ad X ◦ Ad x ( Y ) , Ad x ( Y ) i ,X | Ad x − | = 2 sup Y ∈ g , | Y | =1 h Ad x − ◦ ad X ( Y ) , Ad x − ( Y ) i . Then, we get by induction and for every X ∈ U ( g ) of strictly positive homogeneousdegree: (cid:12)(cid:12) e X d G ( x ) (cid:12)(cid:12) , (cid:12)(cid:12) X d G ( x ) (cid:12)(cid:12) ≤ | ad X | | Ad x | + | Ad x − | q | Ad x | + | Ad x − | ≤ | ad X | d G ( x ) , where, for X ∈ U ( g ), we denote by | ad X | the operator norm of the adjoint actionof U ( g ) on g . This implies condition (i) of Definition 1.1. Last, the inequality | Ad gh | = | Ad g ◦ Ad h | ≤ | Ad g || Ad h | , g, h ∈ G , implies the sub-multiplicativity of d G . (cid:3) We next give further informations regarding the modular weight of a semi-directproduct of groups. These results will be used in Lemmas 2.27, 2.31, Proposition4.20 and Lemma 7.18.
Lemma . Let G j , j = 1 , , be two connected Lie groups with Lie algebras g j ,let R : G → Aut( G ) be an extension homomorphism and consider the associatedsemi-direct product G := G ⋉ R G whose Lie algebra is denoted by g = g ⋉ g .Consider the adjoint actions Ad j : G j → GL ( g j ) and Ad : G → GL ( g ) and definethe mapping Φ : G → Hom( g , g ) , g (cid:2) X Ad g ( X ) − X (cid:3) . Then, in the parametrization G × G → G , ( g , g ) g g , we have the followingbehavior of the modular weight of the semi-direct product: d G ≍ h ( g , g ) d G ( g ) + (cid:0) | Φ( g ) ◦ Ad g | + | Ad g ◦ R g | + | R g − ◦ Φ( g − ) | + | R g − ◦ Ad g − | (cid:1) / i , where we use the same notation for the extension homomorphism and its derivative: g → g , X ddt (cid:12)(cid:12)(cid:12) t =0 R g ( e tX ) , g ∈ G . In particular, we get the following behaviors of the restrictions of d G to the subgroups G and G : d G (cid:12)(cid:12) G ≍ d G + h g (cid:0) | R g | + | R g − | (cid:1) / i d G (cid:12)(cid:12) G ≍ d G + h g (cid:0) | Φ( g ) | + | Φ( g − ) | (cid:1) / i . Moreover, in the case of a direct product, we have the behavior d G × G ≍ d G ⊗ ⊗ d G . Proof.
Fix Euclidean structures on g and on g and induce one on g ⊕ g by declaring that g is orthogonal to g . A direct computation shows that in thevector decomposition g = g ⊕ g , the operator Ad g g takes the following matrixform: Ad g g = (cid:18) Ad g g ) ◦ Ad g Ad g ◦ R g (cid:19) , with inverse given by Ad ( g g ) − = Ad g − R g − ◦ Φ( g − ) R g − ◦ Ad g − ! . But on the finite dimensional vector space
End ( g ), the operator norm of Ad g g isequivalent to its Hilbert-Schmidt norm, and the latter reads (cid:0) | Ad g | + | Φ( g ) ◦ Ad g | + | Ad g ◦ R g | (cid:1) / . Similarly, the operator norm of Ad ( g g ) − is equivalent to (cid:0) | Ad g − | + | R g − ◦ Φ( g − ) | + | R g − ◦ Ad g − | (cid:1) / , proving the first claim. The remaining statements follow immediately. (cid:3) We should mention that the modular function, ∆ G , is also a sub-multiplicativeweight on G . Indeed the multiplicativity property implies that for every X ∈ U ( g )and x ∈ G : (cid:0) e X ∆ G (cid:1) ( x ) = (cid:0) e X ∆ G (cid:1) ( e ) ∆ G ( x ) , (cid:0) X ∆ G (cid:1) ( x ) = (cid:0) X ∆ G (cid:1) ( e ) ∆ G ( x ) . However, this weight will not be of much interest in what follows.The next notion will play a key role to establish density results for our symbolspaces. We assume from now on the Lie group G to be non-compact. Definition . Given two weights µ and ˆ µ , we say that ˆ µ dominates µ ,which we denote by µ ≺ ˆ µ , if lim g →∞ µ ( g )ˆ µ ( g ) = 0 . Remark . For negatively curved K¨ahlerian Lie groups, the modular weighthas the crucial property to dominate the constant weight 1 (see Corollary 2.28 andLemma 2.31). .1. SYMBOL SPACES 13
We now let E be a complex Fr´echet space with topology underlying a countablefamily of semi-norms {k . k j } j ∈ N . Given a weight µ , we first consider the followingspace of E -valued functions on G : B µ ( G, E ) :=(1.2) n F ∈ C ∞ ( G, E ) : ∀ X ∈ U ( g ) , ∀ j ∈ N , ∃ C > k e XF k j ≤ C µ o . When E = C (respectively when µ = 1, respectively when E = C and µ = 1),we denote B µ ( G, E ) by B µ ( G ) (respectively by B ( G, E ), respectively by B ( G )). Weendow the space B µ ( G, E ) with the natural topology associated to the followingsemi-norms:(1.3) k F k j,k,µ := sup X ∈ U k ( g ) sup g ∈ G n k e XF ( g ) k j µ ( g ) | X | k o , j, k ∈ N , where U ( g ) = ∪ k ∈ N U k ( g ) is the filtration described in (0.5) and | . | k is the norm on U k ( g ) defined in (0.7). Note that for X = P | β |≤ k C β X β ∈ U k ( g ), we have k e XF ( g ) k j | X | k ≤ P | β |≤ k | C β |k g X β F ( g ) k j P | β |≤ k | C β | ≤ max | β |≤ k k g X β F ( g ) k j , and hence k F k j,k,µ ≤ max | β |≤ k sup g ∈ G k g X β F ( g ) k j µ ( g ) = max | β |≤ k k g X β F k j, ,µ , (1.4)which shows that the semi-norms (1.3) are well defined on B µ ( G, E ). When E = C (respectively when µ = 1, respectively when E = C and µ = 1), we denote thesemi-norms (1.4) by k . k k,µ , k ∈ N (respectively by k . k j,k , j, k ∈ N , respectively by k . k k , k ∈ N ).The basic properties of the spaces B µ ( G, E ) are established in the next lemma.They are essentially standard but are used all over the text. In particular, in thelast item, we prove that D ( G, E ) is a dense subset of B µ ( G, E ) for the inducedtopology of B ˆ µ ( G, E ), for ˆ µ an arbitrary weight which dominates µ . This fact willbe used in a crucial way for the construction of the oscillatory integral given inDefinition 1.31.Let C b ( G, E ) be the Fr´echet space of E -valued continuous and bounded functionson G . The topology we consider on the latter is the one associated to the semi-norms k F k j := sup g ∈ G k F ( g ) k j , j ∈ N . This space carries an action of G byright-translations. This action is of course isometric but not necessarily stronglycontinuous. Consider therefore its closed subspace C ru ( G, E ) constituted by theright-uniformly continuous functions. Lemma . Let ( G, E ) as above and let µ , ν and ˆ µ be three weights on G . (i) The right regular action R ⋆ of G on C ru ( G, E ) is isometric and stronglycontinuous. (ii) Let C ru ( G, E ) ∞ be the subspace of C ru ( G, E ) of smooth vectors for the rightregular action. Then C ru ( G, E ) ∞ identifies with B ( G, E ) as topologicalvector spaces. In particular, B ( G, E ) is Fr´echet. (iii) The left regular action L ⋆ of G on B ( G, E ) is isometric. (iv) The bilinear map: B µ ( G ) × B ν ( G, E ) → B µν ( G, E ) , ( u, F ) [ g ∈ G u ( g ) F ( g ) ∈ E ] , is continuous. (v) The map B µ ( G, E ) → B ( G, E ) , F µ − F , is an homeomorphism. In particular, the space B µ ( G, E ) is Fr´echet aswell. (vi) For every X ∈ U ( g ) , the associated left invariant differential operator e X acts continuously on B µ ( G, E ) . (vii) If there exists
C > such that µ ≤ C ˆ µ , then B µ ( G, E ) ⊂ B ˆ µ ( G, E ) ,continuously. (viii) Assume that µ ≺ ˆ µ . Then the closure of D ( G, E ) in B ˆ µ ( G, E ) contains B µ ( G, E ) . In particular, the space D ( G, E ) is a dense subset of B µ ( G, E ) for the induced topology of B ˆ µ ( G, E ) . Proof. (i) Recall that G being locally compact and countable at infinity, thespace C b ( G, E ) is Fr´echet (by the same argument as in the proof of [ , Proposition44.1 and Corollary 1]). The subspace C ru ( G, E ) is then closed as a uniform limit ofright-uniformly continuous functions is right-uniformly continuous. Thus C ru ( G, E )endowed with the induced topology is a Fr´echet space as well.Being isometric on C b ( G, E ), the right action is consequently isometric on C ru ( G, E )too. Moreover, for any converging sequence { g n } ⊂ G , with limit g ∈ G , and any F ∈ C ru ( G, E ), we have k ( R ⋆g n − R ⋆g ) F k j = sup g ∈ G k F ( g g n ) − F ( g g ) k j whichtends to zero due to the right-uniform continuity of F . Hence the right regularaction R ⋆ is strongly continuous on C ru ( G, E ).(ii) Note that an element F ∈ C ru ( G, E ) ∞ is such that the map [ g R ⋆g F ] issmooth as a C ru ( G, E )-valued function on G . In particular, for every X ∈ U ( g ), e XF is bounded and smooth. This clearly gives the inclusion C ru ( G, E ) ∞ ⊂ B ( G, E ).Reciprocally, G acts on B ( G, E ) via the right regular representation. Indeed, for all g ∈ G and X ∈ U ( g ), we have e X ◦ R ⋆g = R ⋆g ◦ ^ ( Ad g − X ) , and hence for j, k ∈ N and F ∈ B ( G, E ), we deduce k R ⋆g F k j,k = sup X ∈ U k ( g ) sup g ′ ∈ G n k ^ ( Ad g − X ) F ( g ′ g ) k j | X | k o = sup X ∈ U k ( g ) sup g ′ ∈ G n k ^ ( Ad g − X ) F ( g ′ ) k j | X | k o ≤ | Ad g − | k k F k j,k , where | Ad g | k denotes the operator norm of the adjoint action of G on the (finitedimensional) Banach space ( U k ( g ) , | . | k ). Now we have the inclusion B ( G, E ) ⊂ C ru ( G, E ). Indeed, for F ∈ B ( G ), g ∈ G and for fixed X ∈ g , one observes that (cid:12)(cid:12) F ( g exp( tX )) − F ( g ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z t dd τ ( F ( g exp( τ X ))) d τ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z t e XF ( g exp( τ X )) d τ (cid:12)(cid:12)(cid:12) ≤ | X | k F k | t | , .1. SYMBOL SPACES 15 where k F k = sup X ∈U ( g ) sup g ∈ G | e XF | / | X | . Hence, we obtain the right-uniformcontinuity of F . To show that F ∈ B ( G ) is a differentiable vector for the right-action, we observe that (cid:12)(cid:12)(cid:12) t (cid:0) F ( g exp( tX )) − F ( g ) (cid:1) − (cid:0) e XF (cid:1) ( g ) (cid:12)(cid:12)(cid:12) ≤ Z (cid:12)(cid:12)(cid:12)(cid:0) e XF (cid:1) ( g exp( tτ X )) − (cid:0) e XF (cid:1) ( g ) (cid:12)(cid:12)(cid:12) d τ ≤ Z Z tτ (cid:12)(cid:12)(cid:12)(cid:0) e X F (cid:1) ( g exp( τ ′ X )) (cid:12)(cid:12)(cid:12) d τ ′ d τ ≤ | t | sup g ∈ G n(cid:12)(cid:12) e X F (cid:12)(cid:12) ( g ) o ≤ | X | k F k | t | , which tends to zero together with t . (Here k F k = sup X ∈U ( g ) sup g ∈ G | e XF | / | X | .)This yields differentiability at the unit element. One gets it everywhere else byobserving that(1.5) e X g ( R ⋆g F ) = R ⋆g ( e XF ) , ∀ X ∈ U ( g ) , ∀ g ∈ G , ∀ F ∈ B ( G ) . An induction on the order of derivation implies B ( G ) ⊂ C ru ( G ) ∞ . The E -valuedcase is entirely similar.The assertion concerning the topology follows from the definition of the topologyon smooth vectors [ ].(iii) The fact that G acts isometrically on B ( G, E ) via the left regular represen-tation, follows from k L ⋆g F k j,k = sup X ∈ U k ( g ) sup g ′ ∈ G k e X (cid:0) L ⋆g F (cid:1) ( g ′ ) k j | X | k = sup X ∈ U k ( g ) sup g ′ ∈ G k (cid:0) L ⋆g e XF (cid:1) ( g ′ ) k j | X | k = sup X ∈ U k ( g ) sup g ′ ∈ G k e XF ( g − g ′ ) k j | X | k = sup X ∈ U k ( g ) sup g ′ ∈ G k e XF ( g ′ ) k j | X | k = k F k j,k . (iv) Let u ∈ B µ ( G ) and F ∈ B ν ( G, E ). Using Sweedler’s notation (0.2), wehave for j, k ∈ N : k uF k j,k,µν = sup X ∈ U k ( g ) sup g ∈ G k e X ( uF )( g ) k j µ ( g ) ν ( g ) | X | k ≤ sup X ∈ U k ( g ) sup g ∈ G X ( X ) (cid:12)(cid:12)(cid:0) e X (1) u (cid:1) ( g ) (cid:12)(cid:12) (cid:13)(cid:13)(cid:0) e X (2) F (cid:1) ( g ) (cid:13)(cid:13) j µ ( g ) ν ( g ) | X | k ≤ (cid:16) sup X ∈ U k ( g ) X ( X ) | X (1) | k | X (2) | k | X | k (cid:17) k u k k,µ k F k j,k,ν . Now, for X = P | β |≤ k C β X β ∈ U k ( g ), expanded in the PBW basis (0.4), we have∆ U ( X ) = X | β |≤ k C β X γ ≤ β (cid:18) βγ (cid:19) X γ ⊗ X β − γ , which, with m the dimension of g , implies that(1.6) X ( X ) | X (1) | k | X (2) | k ≤ X | β |≤ k | C β | X γ ≤ β (cid:18) βγ (cid:19) ≤ mk X | β |≤ k | C β | = 2 mk | X | k . Hence we get k uF k j,k,µν ≤ mk k u k k,µ k F k j,k,ν , proving the continuity.(v) Note first that since µ is a weight, for every X ∈ U ( g ) there is a constant C > | e X ( µ − ) | ≤ Cµ − . This easily follows by induction once onehas noticed that for X ∈ g , we have e X ( µ − ) = − ( e Xµ ) µ − . This entails that if µ ∈ B µ ( G ) then µ − ∈ B µ − ( G ), even if µ − may not be a weight in the sense ofDefinition 1.1, as it may not have any sub-multiplicative degree. Then, (v) followsfrom (iv), where the existence of a sub-multiplicative degree is not used in thatproof.(vi) This assertion follows immediately from equation (1.4) and (vii) followseasily from equation (1.3).(viii) Choose an increasing sequence { C n } n ∈ N of relatively compact open sub-sets in G , such that lim n C n = G . Pick 0 ≤ ψ ∈ D ( G ) of L ( G )-norm one anddefine(1.7) e n := Z G ψ ( g ) R ⋆g ( χ n ) d G ( g ) , where χ n denotes the characteristic function of C n . It is clear that e n is an in-creasing family of smooth compactly supported functions, which by Lebesgue dom-inated convergence, converges point-wise to the unit function. Moreover, for all F ∈ B µ ( G, E ) and ˆ µ a weight dominating µ , we have k (1 − e n ) F k j, , ˆ µ = sup g ∈ G n µ ( g ) (cid:0) − e n ( g ) (cid:1) k F ( g ) k j o ≤ k F k j, ,µ sup g ∈ G n µ ( g )ˆ µ ( g ) (cid:0) − e n ( g ) (cid:1)o , which converges to zero when n goes to infinity, since µ/ ˆ µ → g → ∞ andfor fixed g ∈ G , 1 − e n ( g ) decreases to zero when n → ∞ . We need to show thatthe same property holds true for all the semi-norms k . k j,k, ˆ µ , k ≥
1. We use aninduction. First note that if X ∈ g , then we have e Xe n = ddt (cid:12)(cid:12)(cid:12) t =0 R ⋆e tX ( e n ) = ddt (cid:12)(cid:12)(cid:12) t =0 Z G ψ ( g ) R ⋆e tX R ⋆g ( χ n ) d G ( g )= ddt (cid:12)(cid:12)(cid:12) t =0 Z G ψ ( e − tX g ) R ⋆g ( χ n ) d G ( g )= Z G (cid:0) Xψ (cid:1) ( g ) R ⋆g ( χ n ) d G ( g ) . A routine inductive argument then gives(1.8) e Xe n = Z G (cid:0) Xψ (cid:1) ( g ) R ⋆g ( χ n ) d G ( g ) , ∀ X ∈ U ( g ) , which entails k e Xe n k ∞ ≤ k Xψ k < ∞ , ∀ X ∈ U ( g ) . (This means that the sequence { e n } n ∈ N belongs to B ( G ), uniformly in n .) Now,assume that k (1 − e n ) F k j,k, ˆ µ → n → ∞ , for a given k ∈ N , for all F ∈ B µ ( G, E )and all j ∈ N . From the same reasoning as those leading to (1.4) and with X β theelement of the PBW basis of U ( g ) defined in (0.4), we see that k (1 − e n ) F k j,k +1 , ˆ µ ≤ k (1 − e n ) F k j,k, ˆ µ + max | β | = k +1 k g X β (cid:0) (1 − e n ) F (cid:1) k j, , ˆ µ . .1. SYMBOL SPACES 17 We only need to show that the second term in the inequality above goes to zerowhen n → ∞ , as the first does by induction hypothesis. Writing X β = X γ X , with | γ | = k and X ∈ g , by virtue of the Leibniz rule, we get f X γ e X (cid:0) (1 − e n ) F (cid:1) = − f X γ (cid:0) ( e Xe n ) F (cid:1) + f X γ (cid:0) (1 − e n ) e XF (cid:1) . Note that k f X γ (cid:0) (1 − e n ) e XF (cid:1) k j, , ˆ µ ≤ k (1 − e n ) e XF k j,k, ˆ µ , which converges to zero when n → ∞ by induction hypothesis, since e XF ∈ B µ ( G, E )and | γ | = k . Regarding the first term, we have using Sweedler’s notations (0.2) andfor a finite sum: f X γ (cid:0) ( e Xe n ) F (cid:1) = X ( X γ ) (cid:0) g X γ (1) e Xe n (cid:1)(cid:0) g X γ (2) F (cid:1) . Note that R G P X ψ d G = 0 for any P ∈ U ( g ), X ∈ g any ψ ∈ D ( G ). Indeed, thisfollows from an inductive argument starting with Z G Xψ ( g ) d G ( g ) = ddt (cid:12)(cid:12)(cid:12) t =0 Z G L ⋆e tX (cid:0) ψ (cid:1) ( g ) d G ( g ) = ddt (cid:12)(cid:12)(cid:12) t =0 Z G ψ ( g ) d G ( g ) = 0 , for all X ∈ g . Using (1.8), we arrive at f X γ (cid:0) ( e Xe n ) F (cid:1) = X ( X γ ) (cid:16) Z G (cid:0) X γ (1) Xψ (cid:1) ( g ) (cid:0) R ⋆g ( χ n ) − (cid:1)(cid:1) d G ( g ) (cid:17) f X γ (2) F , which converges to zero in the norms k . k j, , ˆ µ , j ∈ N , since it is a finite sum ofterms of the form (1 − e n ) F (with possibly re-defined F ’s in B µ ( G, E ) and ψ ’s in D ( G )). (cid:3) Remark . We stress that as µ ≺ ˆ µ implies that µ ≤ C ˆ µ for some C > G is locally compact), in Lemma 1.8, item (viii) strengthens item (vii). Inparticular, when µ ≺ ˆ µ , B µ ( G, E ) is continuously contained in B ˆ µ ( G, E ). Thisfact is of upmost importance for the continuity of the oscillatory integral, given inDefinition 1.31. Remark . On B ( G, E ), the left regular action is generally not stronglycontinuous and the right regular action is never isometric unless G is Abelian.We now generalize the spaces B µ ( G, E ), by allowing a certain behavior at in-finity of the E -valued functions on G , which is not necessarily uniform with respectto the semi-norm index. We first introduce some more notations. Definition . Let J be a countable set and let µ := { µ j } j ∈ J be an associ-ated family of weights on a Lie group G . We denote by ( L, R ) := { ( L j , R j ) } j ∈ J theassociated family of sub-multiplicative degrees. Given two families of weights µ, ˆ µ ,we say that ˆ µ dominates µ , denoted by µ ≺ ˆ µ , if ˆ µ j dominates µ j for all j ∈ J .The term by term product (respectively tensor product) of two families of weights µ and ν , is denoted by µ.ν (respectively by µ ⊗ ν ).Consider a Fr´echet space E and µ a countable family of weights on G . We thendefine B µ ( G, E ) :=(1.9) n F ∈ C ∞ ( G, E ) : ∀ X ∈ U ( g ) , ∀ j ∈ N , ∃ C > k e XF k j ≤ C µ j o . We endow the latter space with the following set of the semi-norms:(1.10) k F k j,k,µ := sup X ∈ U k ( g ) sup g ∈ G n k e XF ( g ) k j µ j ( g ) | X | k o , j, k ∈ N , As expected, the space B µ ( G, E ) is Fr´echet for the topology induced by the semi-norms (1.10) and most of the properties of Lemma 1.8 remain true. Lemma . Let (cid:0) G, E (cid:1) as above and let µ , ν and ˆ µ be three families of weightson G . (i) The space B µ ( G, E ) is Fr´echet. (ii) Let ( L, R ) be the sub-multiplicative degree of µ . Then, for every g ∈ G theleft-translation L ⋆g defines a continuous map from B µ ( G, E ) to B λ ( G, E ) ,where λ := { λ j } j ∈ N with λ j := µ R j j . (iii) The bilinear map: B µ ( G ) × B ν ( G, E ) → B µ.ν ( G, E ) , ( u, F ) [ g ∈ G u ( g ) F ( g ) ∈ E ] , is continuous. (iv) For every X ∈ U ( g ) , the left invariant differential operator e X acts con-tinuously on B µ ( G, E ) . (v) If for every j ∈ N , there exists C j > such that µ j ≤ C j ˆ µ j , then B µ ( G, E ) ⊂ B ˆ µ ( G, E ) . (vi) Assume that µ ≺ ˆ µ . Then, the closure of D ( G, E ) in B ˆ µ ( G, E ) contains B µ ( G, E ) . In particular, D ( G, E ) is a dense subset of B µ ( G, E ) for theinduced topology of B ˆ µ ( G, E ) . Proof. (i) For each j ∈ N , define k . k ∼ j := P jk =0 k . k k . Clearly, the topolo-gies on E associated with the families of semi-norms {k . k j } j ∈ N and {k . k ∼ j } j ∈ N areequivalent. Thus, we may assume without loss of generality that the family ofsemi-norms {k . k j } j ∈ N is increasing. We start by recalling the standard realizationof the Fr´echet space ( E , {k . k j } j ∈ N ) as a projective limit. One considers the nullspaces V j := { v ∈ E | k v k j = 0 } and form the normed quotient spaces ˙ E j := E /V j .Denoting by E j the Banach completion of the latter, the family of semi-norms be-ing increasing, one gets, for every pair of indices i ≤ j , a natural continuous linearmapping g ji : E j → E i . The Fr´echet space E is then isomorphic to the subspace˜ E of the product space Q j E j constituted by the elements ( x ) ∈ Q j E j such that x i = g ji ( x j ). Within this setting, the subspace ˜ E is endowed with the projectivetopology associated with the family of maps { f j : ˜ E → E j : ( x ) x j } (i.e . the coars-est topology that renders continuous each of the f j ’s— see e.g . [ , pp. 50-52]).Within this context, we then observe that the topology on B µ ( G, E ) ≃ B µ ( G, ˜ E )induced by the semi-norms (1.10) consists of the projective topology associatedwith the mappings φ j : B µ ( G, ˜ E ) → B µ j ( G, E j ) : F f j ◦ F . Next we considera Cauchy sequence { F n } n ∈ N in B µ ( G, ˜ E ). Since every space B µ j ( G, E j ) is Fr´echet,each sequence { f j ◦ F n } n ∈ N converges in B µ j ( G, E j ) to an element denoted by F j .Moreover, for every g ∈ G , one has k g ji ◦ F j ( g ) − F i ( g ) k i = k g ji ◦ F j ( g ) − f i ◦ F n ( g ) + f i ◦ F n ( g ) − F i ( g ) k i ≤ k g ji ◦ (cid:0) F j − f j ◦ F n (cid:1) ( g ) k i + k f i ◦ F n ( g ) − F i ( g ) k i , .1. SYMBOL SPACES 19 which can be rendered as small as we want since every g ji is continuous. Hence g ji ◦ F j = F i which amounts to say that B µ ( G, ˜ E ) is complete.(ii) Let F ∈ B µ ( G, E ) and g ∈ G and set λ j := µ R j j . We have for j, k ∈ N : k L ⋆g F k j,k,λ = sup X ∈ U k ( g ) sup g ′ ∈ G k e X (cid:0) L ⋆g F (cid:1) ( g ′ ) k j µ j ( g ′ ) R j | X | k = sup X ∈ U k ( g ) sup g ′ ∈ G k (cid:0) L ⋆g e XF (cid:1) ( g ′ ) k j µ j ( g ′ ) R j | X | k = sup X ∈ U k ( g ) sup g ′ ∈ G k (cid:0) e XF (cid:1) ( g − g ′ ) k j µ j ( g ′ ) R j | X | k ≤ µ j ( g − ) L j k F k j,k,µ . Items (iii), (iv), (v) and (vi) are proved in the same way as their counterpartsin Lemma 1.8. (cid:3)
Remark . In the same fashion as in Remark 1.9, we see that when µ ≺ ˆ µ ,then B µ ( G, E ) ⊂ B ˆ µ ( G, E ) continuously.In Lemma 1.15, we show on a first example, how the notion of B -spaces forfamilies of weights, naturally appears in the context of non-Abelian Lie group ac-tions. We start by a preliminary result. We fix an Euclidean structure on g , suchthat the basis { X , . . . , X n } (from which we have constructed the PBW basis (0.4))is orthonormal. Lemma . For g ∈ G and k ∈ N , denote by | Ad g | k the operator norm of theadjoint action Ad of G on the finite dimensional Banach space (cid:0) U k ( g ) , | . | k (cid:1) . Then,for each k ∈ N , there exists a constant C k > , such that for all g ∈ G | Ad g | k ≤ C k d G ( g ) k . where d G ∈ C ∞ ( G ) is the modular weight (given in Definition 1.3). Proof.
Note first that for all k ∈ N , there exists a constant ω k > X ∈ U k ( g ) and Y ∈ U k ( g ), we have | X Y | k + k ≤ ω k + k | X | k | Y | k . (1.11)Indeed, observe that if X = X | β |≤ k C β X β ∈ U k ( g ) and Y = X | β |≤ k C β X β ∈ U k ( g ) , we have X Y = X | β |≤ k , | β |≤ k C β C β X | β |≤| β + β | ω β ,β β X β , where the constants ω β ,β β are defined in (0.6). The sub-additivity of the norm | . | k + k then entails that | X Y | k + k ≤ X | β |≤ k , | β |≤ k | C β | | C β | X | β |≤| β + β | | ω β ,β β | . Thus, it leads to defining ω k + k := sup | β + β |≤ k + k X | β |≤| β + β | | ω β ,β β | , and the inequality (1.11) is proved. Next, for X = X | β |≤ k C β X β . . . X β m m ∈ U k ( g ) , we have Ad g ( X ) = X | β |≤ k C β (cid:0) Ad g ( X ) (cid:1) β . . . (cid:0) Ad g ( X m ) (cid:1) β m ∈ U k ( g ) , and thus by the previous considerations, we deduce (cid:12)(cid:12) Ad g ( X ) (cid:12)(cid:12) k ≤ X | β |≤ k | C β | (cid:16) | β | Y j =2 ω j (cid:17) | Ad g ( X ) | β | Ad g ( X ) | β . . . | Ad g ( X m ) | β m . As the restriction of the norm | . | from U ( g ) to g coincides with the ℓ -norm of g within the basis { X , ..., X m } , we deduce for j = 1 , . . . , m and with | Ad g | theoperator norm of Ad g with respect to the Euclidean structure of g chosen: | Ad g ( X j ) | ≤ √ m | Ad g ( X j ) | ≤ √ m | Ad g | | X j | g = √ m | Ad g | , as X j ∈ g belongs to the unit sphere of g for the Euclidean norm | . | g . This implies (cid:12)(cid:12) Ad g ( X ) (cid:12)(cid:12) k ≤ m k/ (cid:16) sup | β |≤ k | β | Y j =2 ω j (cid:17) | Ad g | k , and the result follows from Definition 1.3. (cid:3) Lemma . Let µ be a family of weights on G , with sub-multiplicative degree ( L, R ) . Then the linear map R := h F ∈ C ∞ ( G, E ) (cid:2) g R ⋆g F (cid:3) ∈ C ∞ (cid:0) G, C ∞ ( G, E ) (cid:1)i , is continuous from B µ ( G, E ) to B ν (cid:0) G, B λ ( G, E ) (cid:1) , where ν := { ν j,k } j,k ∈ N and λ := { λ j } j ∈ N with ν j,k := µ R j j d kG and λ j := µ L j j . More precisely, labeling by ( j, k ) ∈ N the semi-norm k . k j,k,λ of B λ ( G, E ) , for each ( j, k, k ′ ) ∈ N , there exists a constant C > , such that for all F ∈ B µ ( G, E ) , wehave kR ( F ) k ( j,k ) ,k ′ ,ν ≤ C k F k j,k + k ′ ,µ . Proof.
Using the relation (1.5), we obtain for X ∈ U k ′ ( g ), F ∈ B µ ( G, E ) and g ∈ G : k e X g R ⋆g ( F ) k j,k,λ = k R ⋆g ( e XF ) k j,k,λ = sup Y ∈ U k ( g ) sup x ∈ G k e Y x R ⋆g (cid:0) e XF (cid:1) ( x ) k j µ L j j ( x ) | Y | k . Moreover, since for any Y ∈ U ( g ) and g ∈ G , we have R ⋆g − e Y R ⋆g = ^ Ad g − Y andsince F ∈ B µ ( G, E ) and µ has sub-multiplicative with degree ( L, R ), we get k e X g R ⋆g ( F ) k j,k,λ = sup Y ∈ U k ( g ) sup x ∈ G k (cid:0) ^ Ad g − Y e XF (cid:1) ( xg ) k j µ L j j ( x ) | Y | k ≤ C k F k j,k + k ′ ,µ | Ad g − | k | X | k ′ sup x ∈ G µ j ( xg ) µ L j j ( x ) ≤ C k F k j,k + k ′ ,µ | Ad g − | k | X | k ′ µ R j j ( g ) , and one concludes using Lemma 1.14. (cid:3) .2. TEMPERED PAIRS 21 In this section, we establish the main technical result of the first part of thismemoir (Proposition 1.29), on which the construction of our oscillatory integral(and thus of our universal deformation formula for Fr´echet algebras) essentiallyrelies. To this aim, we start by introducing the class of tempered Lie groups (Defi-nition 1.17) and the sub-class of tempered pairs (Definition 1.22). In Lemma 1.21,we give simple but important consequences for the modular weight, modular func-tion and Haar measure, when the group is tempered. The rest of this section isdevoted to the proof of Proposition 1.29.
Lemma . Let G be a connected real Lie group and ψ : R m → G be a globaldiffeomorphism. Then the multiplication and inverse operations seen through ψ aretempered functions (in the ordinary sense of R m ) if and only if for every element A ∈ U ( g ) their derivatives along e A are bounded by a function which is polynomialwithin the chart ψ . Proof.
Denote m ( x, y ) = m x ( y ) = ψ − ( ψ ( x ) · ψ ( y )) and ι ( x ) = ψ − ( ψ ( x ) − )the multiplication and inverse of G seen through ψ ∈ Diff( R m , G ). Denote by x e the transportation of the neutral element e of G : ψ ( x e ) := e . Lastly, identifyingnaturally T x e ( R m ) with R m , for every X ∈ R m denote by e X ψx := m x⋆x e ( X ) = ψ − ⋆ ^ (cid:0) ψ ⋆x e X (cid:1) ψ ( x ) , the left invariant vector field corresponding to ψ ⋆x e X ∈ g .Assume m and ι are tempered in the usual sense. Then for X ∈ R m , bydefinition e X ψx = ddt m ( x, x e + tX ) (cid:12)(cid:12)(cid:12) t =0 , which is a linear combination of partial derivatives of m all of them being boundedby some polynomials in x since m is tempered. In the same way, the derivativesof left-invariant vector fields are linear combinations of higher partial derivatives ofcompositions of m with itself in the second variable, which are also bounded by somepolynomials. Hence the left-invariant vector fields are tempered, and consequentlyso are the left-invariant derivatives of m and ι .Conversely, assume m and ι are tempered in the sense of left-invariant vectorfields. We will see that the constant vector fields on R m are linear combinationsof left-invariant vector fields, the coefficients being tempered functions. Indeed, wehave X = (cid:0) m x⋆x e (cid:1) − (cid:0) e X ψx (cid:1) and the matrix elements of that inverse matrix are finitesums and products of the matrix elements of the original one, which are tempered,divided by its determinant. Thus all we have to check is that the inverse of thedeterminant is a tempered function. But 1 / det( m x ⋆xe ) = det( m ι ( x ) ⋆x ) is temperedsince m and ι are. (cid:3) The preceding observation yields us to introduce the following notion:
Definition . A Lie group G is called tempered if there exists a globalcoordinate system ψ : R m → G where the multiplication and inverse operationsare tempered functions. A smooth function f on a tempered Lie group is called a tempered function if f ◦ ψ is tempered. By tempered function, we mean a smooth function whose every derivative is bounded by apolynomial function. These functions are sometimes called “slowly increasing”.
Remark . Every tempered Lie group, being diffeomorphic to an Euclideanspace, is connected and simply connected. Moreover, by arguments similar to thoseof Lemma 1.16, a smooth function f on a tempered Lie group is tempered if andonly if for any X ∈ U ( g ), its derivative along e X is bounded by a polynomial functionwithin the global chart R m → G . Example . For any (simply connected) nilpotent Lie group, the exponen-tial coordinates, g → G : X exp( X ), provides G with a structure of a temperedLie group. Indeed, in the case of a nilpotent Lie group, the Baker-Campbell-Hausdorff series is finite. Remark . Observe also that we can replace in Lemma 1.16 left-invariantdifferential operators by right-invariant one. In fact, for a tempered group, left-invariant vector fields are linear combinations of right-invariant one with temperedcoefficients and vice versa.
Lemma . Let G be a tempered Lie group. Then the modular weight d G andthe modular function ∆ G are tempered. Moreover, in the transported coordinates,every Haar measure on G is a multiple of a Lebesgue measure on R m by a tempereddensity. Proof.
The conjugate action C : G × G → G : ( g, x ) gxg − is a temperedmap when read in the global coordinate system. Therefore, the evaluation of therestriction of its tangent mapping to the first factor G × { e } on the constant section0 ⊕ X , X ∈ g , of T ( G × G ) consists of a tempered mapping: G → g , g C ⋆ ( g,e ) (0 g ⊕ X ) . The latter coincides with g Ad g ( X ). Varying X in the finite dimensional vectorspace g , yields the tempered map Ad : G → End ( g ). This shows that d G is tempered.Transporting the group structure of G to R n by means of the global coordinates,it is clear that any Haar measure on R n (for the transported group law) is absolutelycontinuous with respect to the Lebesgue measure. Let d G ( ξ ) be a left invariant Haarmeasure on G transported to R n . Let also ρ : R n → R be the Radon-Nikodymderivative of d G ( ξ ) with respect to d ξ , the Lebesgue measure on R n . Let ξ e ∈ R n be the transported neutral element of G . By left-invariance of the Haar measured G ( ξ ), we get ρ (cid:0) m ( ξ ′ , ξ ) (cid:1) = ρ ( ξ ) | Jac L ⋆ξ ′ | ( ξ ) , ∀ ξ, ξ ′ ∈ R n , where m ( ., . ) denotes the transported multiplication law on R n and L ⋆ξ stands forthe associated left translation operator on R n . Letting ξ → ξ e , we deduce ρ ( ξ ) = ρ ( ξ e ) | Jac L ⋆ξ | ( ξ e ) , ∀ ξ ∈ R n , and we conclude by Lemma 1.16 using the fact that the multiplication law is tem-pered.Next, we let ι the inversion map of G transported to R n . We have in thetransported coordinates:∆ G ( ξ ) = d G (cid:0) ι ( ξ ) (cid:1) d G ( ξ ) = d G (cid:0) ι ( ξ ) (cid:1) d ξ d ξ d G ( ξ ) = Jac ι ( ξ ) d G (cid:0) ι ( ξ ) (cid:1) d (cid:0) ι ( ξ ) (cid:1) d ξ d G ( ξ ) = Jac ι ( ξ ) ρ (cid:0) ι ( ξ ) (cid:1) ρ ( ξ ) , and we conclude using what precedes and the temperedness of the inversion mapon G . (cid:3) .2. TEMPERED PAIRS 23 We now consider the data of a pair (
G, S ) where G is a connected real Liegroup with real Lie algebra g and S is a real-valued smooth function on G . Definition . The pair (
G, S ) is called tempered if the following twoproperties are satisfied:(i) The map(1.12) φ : G → g ⋆ , x h g → R , X (cid:0) e X. S (cid:1) ( x ) i , is a global diffeomorphism.(ii) The inverse map φ − : g ⋆ ≃ R m → G endows G with the structure of atempered Lie group. Remark . Within the above situation, the function S is itself automat-ically tempered. Indeed, in the proof of Lemma 1.16, we have seen that direc-tional derivatives in the coordinate system φ are expressed as linear combinationsof left-invariant vector fields with tempered coefficients. Hence, within a basis { X j } j =1 ,...,N of g , denoting x j = (cid:0) f X j S (cid:1) ( x ), we have ∂ x j S ( x ) = P k m kj ( x ) x k wherethe matrix ( m kj ) j,k have tempered entries. This implies that the partial derivatives(of every strictly positive order) of S are tempered. In spherical coordinates ( r, θ )(associated to the x j ’s) one observes that ∂ r S ( r, θ ) = R ( θ ) k ∂ x k S where θ belongs tothe unit sphere S N − and where R ( θ ) is a (rotation) matrix that smoothly dependson θ . Hence: | S ( x ) | = (cid:12)(cid:12)(cid:12) C + Z rr ∂ ρ S ( ρ, θ )d ρ (cid:12)(cid:12)(cid:12) ≤ | C | + Z rr (cid:12)(cid:12) R ( θ ) k (cid:12)(cid:12) ρ n k d ρ , which for large x is smaller than a multiple of some positive power of r . Thereforethe function S has polynomial growth too.Given a tempered pair ( G, S ), with g the Lie algebra of G , we now consider avector space decomposition:(1.13) g = N M n =0 V n , and for every n = 0 , . . . , N , an ordered basis { e nj } j =1 ,..., dim( V n ) of V n . We get globalcoordinates on G :(1.14) x jn := (cid:0)f e nj .S (cid:1) ( x ) , n = 0 , . . . , N , j = 1 , . . . , dim( V n ) . We choose a scalar product on each V n and let | . | n be the associated Euclidean norm.Given an element A ∈ U ( g ), we let e A ∗ be the formal adjoint of the left-invariantdifferential operator e A , with respect to the inner product of L ( G ). We make theobvious observation that e A ∗ is still left-invariant. Indeed, for ψ, ϕ ∈ C ∞ c ( G ) and g ∈ G , since L ⋆g is unitary on L ( G ), we have h L ⋆g e A ∗ ψ, ϕ i = h ψ, e AL ⋆g − ϕ i = h ψ, L ⋆g − e Aϕ i = h e A ∗ L ⋆g ψ, ϕ i . Moreover, we make the following requirement of compatibility of the adjoint mapon L ( G ) with respect to the ordered decomposition (1.13). Namely, denoting forevery n ∈ { , ..., N } : V ( n ) := n M k =0 V k , (1.15) we assume:(1.16) ∀ n = 0 , . . . , N, ∀ A ∈ U ( V n ) , ∃ B ∈ U ( V ( n ) ) such that e A ∗ = e B , where the space U ( V ( n ) ) is the subalgebra of U ( g ) generated by V ( n ) , as defined in(0.8). We now pass to regularity assumptions regarding the function S . Definition . Set(1.17) E := exp { iS } . A tempered pair (
G, S ) is called admissible , if there exists a decomposition (1.13)with associated coordinate system (1.14), such that for every n = 0 , . . . , N , thereexists an element X n ∈ U ( V n ) ⊂ U ( g ) whose associated multiplier α n , defined as e X n E =: α n E , satisfies the following properties:(i) There exist C n > ρ n > | α n | ≥ C n (cid:0) | x n | ρ n n (cid:1) , where x n := ( x jn ) j =1 ,..., dim( V n ) .(ii) For all n = 0 , . . . , N , there exists a tempered function µ n ∈ C ∞ ( G, R ∗ + )such that:(ii.1) For every A ∈ U ( V ( n ) ) ⊂ U ( g ) there exists C A > (cid:12)(cid:12) e A α n (cid:12)(cid:12) ≤ C A | α n | µ n . (ii.2) n µ n is independent of the variables { x jr } j =1 ,..., dim( V r ) , for all r ≤ n : ∂µ n ∂x jr = 0 , ∀ r ≤ n , ∀ j = 1 , . . . , dim( V r ) . We now give a sequence of Lemmas allowing the construction of a sequenceof differential operator { D j } j ∈ N such that, for µ a family of tempered weights, D j sends continuously B µ j ( G, E j ) to L ( G, E j ), where E j is the semi-normed space( E , k . k j ) and which is such that D ∗ j E = E . Thus is all that we need to constructour oscillatory integral. We start with a preliminary result, which gives an upperbound for powers of derivatives of the inverse of a multiplier, in the context ofadmissible tempered pairs. Lemma . Fix n = 0 , . . . , N . Let α ∈ C ∞ ( G ) be non-vanishing and ≤ µ ∈ C ∞ ( G ) such that for every A ∈ U ( V ( n ) ) there exists C > with | e Aα | ≤ C µ | α | .Fixing X ∈ U ( V ( n ) ) , a monomial of homogeneous degree M ∈ N , we consider thedifferential operator D X,α : C ∞ ( G ) → C ∞ ( G ) , Φ e X (cid:16) Φ α (cid:17) . Then, for every r ∈ N , there exist an element X ′ ∈ U ( g ) of maximal homogeneousdegree bounded by rM and a constant C > such that for every Φ ∈ C ∞ ( G ) wehave: (cid:12)(cid:12) D rX,α Φ (cid:12)(cid:12) ≤ C µ r M | α | r (cid:12)(cid:12) e X ′ Φ (cid:12)(cid:12) . .2. TEMPERED PAIRS 25 Proof.
We start by recalling Fa`a di Bruno’s formula:d r d t r (cid:16) f (cid:17) = 1 f X ~M C r~M r Y j =1 (cid:16) f ( j ) f (cid:17) M j , f ∈ C ∞ ( R ) , where ~M = ( M , . . . , M r ) runs along partitions of r (i.e . r = P rj =1 jM j ) and where C r~M is some combinatorial coefficient. Within Sweedler’s notations (0.2), Fa`a diBruno formula then yields for Φ ∈ C ∞ ( G ): D X,α
Φ = X ( X ) (cid:16) e X (1) α (cid:17) (cid:16) e X (2) Φ (cid:17) = X ( X ) α X Y (cid:16) e X j αα (cid:17) M j (cid:16) e X (2) Φ (cid:17) , where the second sum and product run over partitions of M (1) := deg( X (1) ) ≤ M and where the element X j is of homogeneous degree j = 1 , . . . , r and contains thecombinatorial coefficients of Fa`a di Bruno’s formula. Of course, we also have that X (1) , X (2) and X j all belong to U ( V ( n ) ). Thus, | e X j α | ≤ C ( X ) µ | α | , the estimationis satisfied for r = 1. For r = 2, we observe: D X,α
Φ = X ( X ) (cid:16) e X (1) α (cid:17) e X (2) (cid:16) X ( X ) e X (1) α e X (2) Φ (cid:17) = X ( X ) , ( X (2) ) (cid:16) e X (1) α (cid:17) X ( X ) (cid:16) e X (21) e X (1) α (cid:17) (cid:16) e X (22) e X (2) Φ (cid:17) . Fa`a di Bruno’ s formula for α then yields the assertion for r = 2. Iterating thisprocedure, we get that(1.18) D rX,α Φ = X ( X ) r Y j =1 (cid:16) e X ( j ) α (cid:17) (cid:16) e X ′ Φ (cid:17) , for some elements X ( j ) , X ′ ∈ U ( V ( n ) ) where the maximal homogeneous degree of X ( j ) is bounded by jM . (We have absorbed the combinatorial coefficients of Fa`adi Bruno’ s formula in X ( j ) and X ′ .) Therefore, Fa`a di Bruno’s formula yields forevery j = 1 , . . . , r : (cid:12)(cid:12)(cid:12) e X ( j ) α (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) α X Y (cid:16) e X ( j ) k .αα (cid:17) deg( X ( j ) k ) (cid:12)(cid:12)(cid:12) ≤ C µ P k deg( X ( j ) k ) | α |≤ C µ deg( X ( j ) ) | α | ≤ C µ jM | α | . Therefore since r ( r +1)2 ≤ r and µ ≥
1, we get the (rough) estimation: (cid:12)(cid:12) D rX,α Φ (cid:12)(cid:12) ≤ C X ( X ) r Y j =1 | α | µ jM (cid:12)(cid:12) e X ′ Φ (cid:12)(cid:12) ≤ C ′ X ( X ) | α | r µ r M (cid:12)(cid:12) e X ′ Φ (cid:12)(cid:12) , which delivers the proof. (cid:3) We now fix an admissible tempered pair (
G, S ) and for all n = 0 , . . . , N , welet X n ∈ U ( V n ) as given in Definition 1.24 and we let α n , µ n ∈ C ∞ ( G ) be the associated multiplier and tempered function. Accordingly to the previous notations,we introduce the differential operators: D n := D X ∗ n ,α n : C ∞ ( G ) → C ∞ ( G ) , Φ e X ∗ n (cid:16) Φ α n (cid:17) . (1.19)Recall that by assumption, there exists Y n ∈ U ( V ( n ) ) such that e X ∗ n = e Y n and thus,we can apply Lemma 1.25 to these operators. For every r n ∈ N , accordingly to theexpression (1.18), we write for Φ ∈ C ∞ ( G ): D r n n Φ = X ( X n ) r n Y j =1 (cid:16) e X ( j ) n α n (cid:17) (cid:16) e X ′ n Φ (cid:17) , where X ( j ) n ∈ U ( V ( n ) ) and its homogeneous degree is bounded by jM n , with M n the maximal homogeneous degree of X n and where the one of X ′ n is bounded by r n M n . Setting(1.20) Ψ n = r n Y j =1 (cid:16) e X ( j ) n α n (cid:17) , we then write (abusively since in fact it is a finite sum of such terms):(1.21) D r n n =: Ψ n e X ′ n . Given a N + 1-tuple of integers ~r = ( r , . . . , r N ), we will be led to consider theoperator(1.22) D ~r := D r ,...,r N := D r D r . . . D r N N . Using the iterated Sweedler’s notation (0.3), a recursive use of (1.21) yields D ~r = X ( X ′ ) X ( X ′ ) · · · X ( X ′ N − ) X ( X ′ N − ) (cid:2) Ψ (cid:3)(cid:2) ( e X ′ ) (1) Ψ (cid:3)(cid:2) ( e X ′ ) (21) ( e X ′ ) (1) Ψ (cid:3)(cid:2) ( e X ′ ) (221) ( e X ′ ) (21) ( e X ′ ) (1) Ψ (cid:3) . . . (cid:2) ( e X ′ ) (22 ... ( e X ′ ) (2 ... . . . ( e X ′ N − ) (21) ( e X ′ N − ) (1) Ψ N − (cid:3)(cid:2) ( e X ′ ) (222 ... ( e X ′ ) (22 ... . . . ( e X ′ N − ) (221) ( e X ′ N − ) (21) ( e X ′ N − ) (1) Ψ N (cid:3) ( e X ′ ) (222 ... ( e X ′ ) (22 ... . . . ( e X ′ N − ) (222) ( e X ′ N − ) (22) ( e X ′ N − ) (2) e X ′ N . Set then, Ψ , := ( e X ′ ) (1) Ψ (1.23) Ψ , , := ( e X ′ ) (21) ( e X ′ ) (1) Ψ Ψ , , , := ( e X ′ ) (221) ( e X ′ ) (21) ( e X ′ ) (1) Ψ ...Ψ N − ,..., , := ( e X ′ ) (22 ... ( e X ′ ) (2 ... . . . ( e X ′ N − ) (21) ( e X ′ N − ) (1) Ψ N − Ψ N,N − ,..., , := ( e X ′ ) (222 ... ( e X ′ ) (22 ... . . . ( e X ′ N − ) (21) ( e X ′ N − ) (1) Ψ N , and X ′ N,..., := ( X ′ ) (222 ... ( X ′ ) (22 ... . . . ( X ′ N − ) (222) ( X ′ N − ) (22) ( X ′ N − ) (2) X ′ N , .2. TEMPERED PAIRS 27 in terms of which we have (with the same abuse of notations as in (1.22) above):(1.24) D ~r = Ψ Ψ , Ψ , , Ψ , , , . . . Ψ N − ,..., , Ψ N,..., e X ′ N,N − ,..., , . Lemma . Fix n = 0 , . . . , N and let α ∈ C ∞ ( G ) and µ ∈ C ∞ ( G, R ∗ + ) satisfying the hypothesis of Lemma 1.25. For j = 1 , . . . , r and r ∈ N ∗ , fix also X ( j ) ∈ U ( V ( n ) ) and define Ψ := r Y j =1 (cid:16) e X ( j ) α (cid:17) , where deg( X ( j ) ) ≤ jM , for a given M ∈ N ∗ . Consider a monomial Y ∈ U ( V ( n ) ) ,then we have e Y Ψ = X ( Y ) r Y j =1 (cid:16) e Y ( j ) α (cid:17) with deg( Y ( j ) ) ≤ jM + deg( Y ) , and moreover there exists C > such that (cid:12)(cid:12) e Y Ψ (cid:12)(cid:12) ≤ C µ r M + r deg( Y ) | α | r . Proof.
The equality is immediate. Regarding the inequality, we first notethat by virtue of Fa`a di Bruno’s formula, we have for a finite sum: e Y ( j ) α = 1 α X deg( Y ( j ) ) Y k =1 (cid:16) e Y ( j ) k αα (cid:17) deg( Y ( j ) k ) . Hence (cid:12)(cid:12)(cid:12) e Y ( j ) α (cid:12)(cid:12)(cid:12) ≤ C µ P k deg( Y ( j ) k ) | α | ≤ C µ deg( Y ( j ) ) | α | ≤ C µ jM +deg( Y ) | α | . We then conclude as in the proof of Lemma 1.25. (cid:3)
From the lemmas above, we deduce an estimate for the ‘coefficient functions’appearing in the expression of the differential operator D ~r in (1.24). Corollary . Let ( G, S ) be an admissible tempered pair with decomposi-tion g = L Nn =0 V n and accordingly to Definition 1.24, for n = 0 , . . . , N , we let ( X n , α n , µ n ) ∈ U ( V n ) × C ∞ ( G ) × C ∞ ( G ) be the associated differential operator,multiplier and tempered function. Then, for k = 0 , . . . , N and r k ∈ N ∗ , with Ψ k,..., ∈ C ∞ ( G ) defined in (1.23) , we have | Ψ k,..., | ≤ C k µ r k M k + r k P k − j =0 r j M j k | α k | r k , for some finite non-negative constant C k and where M n := deg( X n ) , n = 0 , . . . , N . Proof.
Observe that Ψ k,..., = k − Y j =0 ( f X ′ j ) (2 ... Ψ k , where Ψ k is defined in (1.20). Since ( X ′ j ) (2 ... ∈ U ( V ( k ) ) with homogeneous degreeof is bounded by r j M j for every j = 0 , . . . , k −
1, the estimate we need follows fromLemma 1.26. (cid:3)
We can now state the main technical results of this chapter.
Proposition . Let ( G, S ) be an admissible tempered pair and let µ be atempered weight. Then, there exists ~r = ( r , . . . , r N ) ∈ N N +1 such that for everyelement F ∈ B µ ( G ) , the function D ~r F belongs to L ( G ) . More precisely, thereexist a finite constant C > and K ∈ N with K ≤ P Nk =0 r k M k and M k = deg( X k ) (with X k ∈ U M k ( g ) as given in Definition 1.24), such that for all F ∈ B µ ( G ) , wehave: k D ~r F k ≤ C sup X ∈ U K ( g ) sup g ∈ G n (cid:12)(cid:12) e X F ( x ) (cid:12)(cid:12) µ ( g ) | X | K o = C k F k K,µ . Proof.
By Lemma 1.21, in the coordinates (1.14), the Radon-Nikodym de-rivative of the left Haar measure on G with respect to the Lebesgue measure on g ⋆ , is bounded by a polynomial in { x jn , j = 1 , . . . , dim( V n ) , n = 0 , . . . , N } . Bythe assumption of temperedness of the weight µ , the latter is also bounded by apolynomial in the same coordinates. Now, observe from (1.24), that we have forany ~r = ( r , . . . , r N ) and for K = deg( X ′ N,..., ) ≤ P Nk =0 r k M k : | D ~r F | ≤ | Ψ | | Ψ , | | Ψ , , | . . . | Ψ N,..., | (cid:12)(cid:12) e X ′ N,..., F (cid:12)(cid:12) ≤ | Ψ | | Ψ , | | Ψ , , | . . . | Ψ N,..., | µ | X ′ N,..., | K k F k K,µ . (1.25)This will gives the estimate, if we prove that the function in front of k F k K,µ in(1.25) is integrable for a suitable choice of ~r ∈ N N +1 . We prove a stronger result,namely that given ~R = ( R , . . . , R N ) ∈ N N +1 , there exists ~r = ( r , . . . , r N ) ∈ N N +1 such that the associated functions Ψ k,..., (which depend on ~r ) satisfy: | Ψ ( x ) | | Ψ , ( x ) | | Ψ , , ( x ) | . . . | Ψ N,..., ( x ) | ≤ C (1 + | x | ) R . . . (1 + | x N | ) R N . From Corollary 1.27 and writing r k M k + r k P k − j =0 r j M j = r k P kj =0 r j M j , we obtainthe following estimation: | Ψ | | Ψ , | | Ψ , , | . . . | Ψ N,..., | ≤ C N Y k =0 µ r k P kj =0 r j M j k α r k k . Moreover, by assumption of temperedness, see Definition 1.24 (ii.1), there exist ρ , . . . , ρ N > | Ψ ( x ) | | Ψ , ( x ) | | Ψ , , ( x ) | . . . | Ψ N,..., ( x ) | ≤ C N Y k =0 µ k ( x ) r k P kj =0 r j M j (1 + | x k | ) ρ k r k . From the hypothesis of Definition 1.24 (ii.2), we deduce that the element µ N isconstant. Indicating the variable dependence into parentheses, one also has µ N − = µ N − ( x N ) , µ N − = µ N − ( x N − , x N ) , . . .µ = µ ( x , . . . , x N ) , µ = µ ( x , x , . . . , x N ) . Denoting by m n , n = 0 , . . . , N , the degree of a polynomial function that, in thevariables (1.14), dominates the tempered function µ n , we obtain the sufficient con-ditions: r ρ ≥ R and ρ n r n − n − X k =0 (cid:16) m k r k k X j =0 r j M j (cid:17) ≥ R n , n = 1 , . . . , N , which are always achievable. (cid:3) .3. AN OSCILLATORY INTEGRAL FOR ADMISSIBLE TEMPERED PAIRS 29 Let now E be a complex Fr´echet space, with topology associated with a count-able family of semi-norms {k . k j } j ∈ N . An immediate modification of its proof, leadus to the following version of Proposition 1.28 (the only difference with the formeris that now the index K ∈ N may depend on j via the order of the tempered weight µ j ). Proposition . Let ( G, S ) be an admissible tempered pair, E be a complexFr´echet space and let µ be a family of tempered weights. Then for all j ∈ N , thereexist ~r j ∈ N N +1 , C j > and k j ∈ N , such that for every element F ∈ B µ ( G, E ) , wehave Z G k D ~r j F ( g ) k j d G ( g ) ≤ C j sup X ∈ U Kj ( g ) sup g ∈ G n k e X F ( x ) k j µ j ( g ) | X | k j o =: C j k F k j,k j ,µ . Our notion of oscillatory integral is an immediate consequence of Proposition1.29 together with the following immediate but essential observation:
Proposition . Let ( G, S ) be an admissible tempered pair, let E be thefunction on G defined in (1.17) and, for ~r ∈ N N +1 , let D ~r be the differentialoperator given in (1.22) . Then, with D ∗ ~r the formal adjoint of D ~r with respect tothe inner product of L ( G ) , we have: D ∗ ~r E = E , ∀ ~r ∈ N N +1 , Proof.
By construction, we have D ~r = D r D r . . . D r N N , where the operators D n , n = 0 , . . . , N , are given in (1.19). Hence it suffices toprove the identity above for ~r = (0 , . . . , , , , . . . , D ∗ n = 1 α n e X n and e X n E = α n E . This completes the proof. (cid:3)
Definition . Let (
G, S ) be an admissible tempered pair, µ a temperedweight, m an element of B µ ( G ) and µ, ˆ µ two families of tempered weights such that µ ≺ ˆ µ (hence µ.µ ≺ ˆ µ.µ ). Associated to the family of weights ˆ µ.µ , let ~r j ∈ N N +1 , j ∈ N , as given in Proposition 1.29 and let D ~r j be the differential operators givenin (1.22). Performing integrations by parts, the Dunford-Pettis theorem [ ] yieldsa B ˆ µ ( G, E )-continuous mapping D ( G, E ) → E , F Z G mE F = Z G E D ~r j (cid:0) m F (cid:1) . Then by Lemma 1.12 (v) (and Remark 1.13), the latter extends to the followingcontinuous linear mapping: ^ Z G m E : B µ ( G, E ) → E , that we refer to as an oscillatory integral . Our next aim is to prove that the oscillatory integral on B µ ( G, E ), does notdepend on the choices made. So let µ , µ and ˆ µ as in Definition 1.31. Fix F ∈B µ ( G, E ) and chose a sequence { F n } n ∈ N of elements of D ( G, E ) converging to F for the topology of B ˆ µ ( G, E ). By definition of the oscillatory integral and undoingthe integrations by parts at the level of smooth compactly supported E -valuedfunctions, we first observe that by continuity, we have: ^ Z G m E (cid:0) F (cid:1) = ^ Z G m E (cid:0) lim n →∞ F n (cid:1) = lim n →∞ ^ Z G m E (cid:0) F n (cid:1) = lim n →∞ Z G m ( g ) E ( g ) F n ( g ) d G ( g ) , where the first limit is in B ˆ µ ( G, E ) and the last two are in E . Then, the estimateof Proposition 1.29 immediately implies that the limit above is independent of theapproximation sequence { F n } n ∈ N chosen. This shows that the oscillatory integraldoes not depend on the differential operators in D ~r j used to define the extension (inthe topology of B ˆ µ ( G, E )) of the oscillatory integral from D ( G, E ) to B µ ( G, E ). Last,to see that the oscillatory integral mapping is also independent of the choice of thefamily of dominant weights ˆ µ chosen, it suffices to remark that the approximationsequence constructed in the proof of Lemma 1.8 (viii) can be used for any familyˆ µ such that µ j ≺ ˆ µ j . Of course this will hold provided that we can always finddominant weights. This is certainly the case if there exists a weight dominating theconstant weight 1. Thus we have proved: Proposition . Let ( G, S ) be an admissible tempered pair, E be a complexFr´echet space, µ be tempered weights, m ∈ B µ ( G ) and µ be a family of temperedweights. Assuming that there exists a tempered weight µ c which dominates theconstant weight , then the oscillatory integral mapping ^ Z G m E : B µ ( G, E ) → E , does not depend on the choice of the integers ~r j ∈ N N +1 and on the family ofdominant weights ˆ µ given in Definition 1.31. Moreover, given F ∈ B µ ( G, E ) , wehave ^ Z G m E (cid:0) F (cid:1) = lim n →∞ Z G m ( g ) E ( g ) F n ( g ) d G ( g ) , where { F n } n ∈ N is an arbitrary sequence in D ( G, E ) , converging to F in the topologyof B ˆ µ ( G, E ) , for an arbitrary sequence of weights ˆ µ , which dominates µ . Remark . Note that Proposition 1.32 does not assert that the oscillatoryintegral on B µ ( G, E ) is the unique continuous extension of its restriction to D ( G, E ).We observe that the existence of a tempered weight that dominates the constantweight 1 implies that every weight is dominated, which is crucial for the constructionof the oscillatory integral, as observed above. This leads us to introduce the notionof tameness below. We will see that this property holds for negatively curvedK¨ahlerian groups, where we can simply use the modular weight (see Corollary 2.28and Lemma 2.31). .3. AN OSCILLATORY INTEGRAL FOR ADMISSIBLE TEMPERED PAIRS 31 Definition . A tempered Lie group G , with associated diffeomorphism φ : G → g ⋆ is called tame if there exist an Euclidean norm | . | on g ⋆ , a temperedweight µ φ and two positive constants C, ρ such that C (cid:0) | φ | (cid:1) ρ ≤ µ φ . Remark . Observe that when a tempered Lie group G is tame, then thereexists a tempered weight that dominates the constant weight 1. Indeed, 1 ≺ µ φ .This is the main reason for introducing this notion.In the constant family case, i.e . for B µ ( G, E ), we can express the oscillatoryintegral as an absolutely convergent one for each semi-norm k . k j : ^ Z G m E (cid:0) F (cid:1) = Z G E D ~r (cid:0) m F (cid:1) , ∀ F ∈ B µ ( G, E ) , where the label ~r ∈ N N +1 of the differential operator D ~r is given by Proposition1.28 and does not depend on the index j ∈ N which labels the semi-norms definingthe topology of E nor on the element F ∈ B µ ( G, E ). However, we cannot accessto such a formula in the case of B µ ( G, E ), since in this case the label ~r ∈ N N +1 depends on the index j ∈ N . However, and from Proposition 1.29, we have thefollowing weaker statement: Proposition . Let ( G, S ) an admissible and tame tempered pair, E aFr´echet space, µ, µ be tempered weights and m ∈ B µ ( G ) . Then for every j ∈ N ,there exists ~r j ∈ N N +1 , such that for all F ∈ B µ ( G, E ) , we have ^ Z G m E (cid:0) F (cid:1) = Z G E D ~r j (cid:0) m F (cid:1) , where the right hand side is an absolutely convergent integral for the semi-norm k . k j . We close this section with a natural result on the compatibility of the oscillatoryintegral with continuous linear maps between Fr´echet spaces.
Lemma . Let ( G, S ) be an admissible and tame tempered pair, ( E , {k . k j } ) and ( F , {k . k ′ j } ) two Fr´echet spaces and T : E → F a continuous linear map. Definethe map ˆ T from C ( G, E ) to C ( G, F ) , by setting (cid:0) ˆ T F (cid:1) ( g ) := T (cid:0) F ( g ) (cid:1) . Then, for any family µ of tempered weights, there exists another family of temperedweights ν , such that ˆ T is continuous from B µ ( G, E ) to B ν ( G, F ) . Moreover for any m ∈ B µ ( G ) , with µ another tempered weight, we have T (cid:16) ^ Z G m E (cid:0) F (cid:1)(cid:17) = ^ Z G m E (cid:0) ˆ T F (cid:1) . (1.26) Proof.
By continuity of T , for all j ∈ N there exist l ( j ) ∈ N and C j > a ∈ E , we have k T ( a ) k ′ j ≤ C j k a k l ( j ) . This immediately implies thecontinuity of ˆ T . Indeed, setting ν j := µ l ( j ) , then for F ∈ B ν ( G, E ) and j, k ∈ N , we have k ˆ T F k ′ j,k,ν = sup X ∈ U k ( g ) sup g ∈ G k e XT (cid:0) F ( g ) (cid:1) k ′ j µ l ( j ) | X | k = sup X ∈ U k ( g ) sup g ∈ G k T (cid:0) e XF ( g ) (cid:1) k ′ j µ l ( j ) | X | k ≤ C j sup X ∈ U k ( g ) sup g ∈ G k (cid:0) e XF ( g ) (cid:1) k l ( j ) µ l ( j ) | X | k = C j k F k l ( j ) ,k,µ . Repeating the arguments for B µ.µ φ ( G, E ) ( µ φ is the tempered weight associated totameness) instead of B µ ( G, E ), we see that both sides of (1.26) define continuouslinear maps from B µ.µ φ ( G, E ) to F . Moreover, it is easy to see that they coincideon D ( G, E ) and thus they coincide on the closure of D ( G, E ) inside B µ.µ φ ( G, E ),which contains B µ ( G, E ) by Lemma 1.12 (vi), as µ.µ φ dominates µ . (cid:3) The aim of this section is to prove a Fubini type result for the oscillatory integralon a semi-direct product of tempered pairs. We start with following observation:
Lemma . Let E be a Fr´echet space, G , G be two Lie groups with Lie alge-bras g , g and R ∈ Hom (cid:0) G , Aut ( G ) (cid:1) be an extension homomorphism. Consider µ , a family of weights on the semi-direct product G ⋉ R G , with sub-multiplicativedegree ( L, R ) . Set also µ and µ for its restrictions to the subgroups G and G and let d be the restriction of the modular weight (cf . Definition 1.3) of G ⋉ R G to G . Then the map C ∞ ( G ⋉ R G , E ) → C ∞ (cid:0) G , C ∞ ( G , E ) (cid:1) ,F ˆ F := (cid:2) g ∈ G [ g ∈ G F ( g g )] (cid:3) , (1.27) sends continuously B µ ( G ⋉ R G , E ) to B ν (cid:0) G , B λ ( G , E ) (cid:1) , where ν := { ν j,k } j,k ∈ N and λ := { λ j } j ∈ N with ν j,k := µ R j ,j d k , λ j := µ L j ,j . Proof.
First, observe that for g ∈ G ⋉ R G with g = g g , g ∈ G , g ∈ G , F ∈ C ∞ ( G ⋉ R G ) and for X ∈ g , X ∈ g , we have f X .g ˆ F ( g , g ) = f X .g F ( g ) , f X .g ˆ F ( g , g ) = ^ R g − ( X ) .g F ( g ) , (1.28)where we use the same notation for the extension homomorphism and its derivative: g → g , X ddt (cid:12)(cid:12)(cid:12) t =0 R g ( e tX ) , g ∈ G . From this, it follows that the restriction of a weight on G ⋉ R G to G or G is still a weight on G or G . Indeed, given µ a weight on G ⋉ R G , call µ i , i = 1 ,
2, its restriction to the subgroup G i and given X ∈ U ( g i ) call X i its image in U ( g ⋉ g ). Then, Equation (1.28) yields e Xµ i = (cid:0) e X i µ (cid:1) i , i = 1 ,
2, which togetherwith ( µ ∨ ) i = ( µ i ) ∨ (where µ ∨ ( g ) := µ ( g − )) implies that the first condition ofDefinition 1.1 is satisfied. Sub-multiplicativity at the level of each subgroups G i , i = 1 ,
2, follows from sub-multiplicativity at the level of G ⋉ R G (with the samesub-multiplicativity degree). .4. A FUBINI THEOREM FOR SEMI-DIRECT PRODUCTS 33 Moreover, (1.28) also implies that for F ∈ B µ ( G ⋉ R G , E ), X ∈ U k ( g ), X ∈ U k ( g ) and k , k , j ∈ N , we have for g g ∈ G ⋉ R G : k f X .g f X .g ˆ F ( g , g ) k j = k (cid:0) f X ^ R g − ( X ) F (cid:1) ( g g ) k j ≤ C ( k , k ) | X | k | X | k | R g − | k sup Y ∈ U k k ( g ⋉ g ) k e Y F ( g g ) k j | Y | k + k ≤ C ( k , k ) | X | k | X | k | R g − | k µ j ( g g ) k F k j,k + k ,µ ≤ C ′ ( k , k ) | X | k | X | k d ( g ) k µ j ( g ) R j µ j ( g ) L j k F k j,k + k ,µ , by Lemma 1.14, since for g ∈ G , R g coincides with the restriction of Ad g to g .Thus, labeling by ( j, k ) ∈ N the semi-norms k . k j,k ,λ of B λ ( G , E ), we finally get: k ˆ F k ( j,k ) ,k ,ν ≤ C ′ ( k , k ) k F k j,k + k ,µ , which completes the proof. (cid:3) Now, assume that the groups G and G come from admissible tempered pairs( G , S ) and ( G , S ). Parametrizing g = g g ∈ G ⋉ R G with g i ∈ G i , i = 1 , S : G ⋉ R G → R , g g S ( g ) + S ( g ) , and using the notation (1.17), we set accordingly E ( g g ) := exp { iS ( g g ) } = E ( g ) E ( g ) . For g ∈ G , we let | R g | be the operator norm of R g ∈ End ( g ). Assume furtherthat the map G → R + , g
7→ | R g | is tempered. Then, by Lemma 1.5 and Lemma1.21, we deduce that d , the restriction of the modular weight on G ⋉ R G to G ,is tempered too. Thus for m ∈ B µ ( G ⋉ R G ) with µ a tempered weight, and withˆ m the associated function on G × G as constructed in (1.27), Lemma 1.38 showsthat the map B µ ( G ⋉ R G , E ) → E , F ^ Z G E (cid:16) ^ Z G E (cid:0) ˆ m ˆ F (cid:1)(cid:17) , is well defined as a continuous linear map. Thus under these circumstances, thismap could be used as a definition for the oscillatory integral on the semi-directproduct G ⋉ R G . Moreover, when the pair (cid:0) G ⋉ R G , S (cid:1) is also tempered andadmissible and when the extension homomorphism preserves the Haar measure d G ,then the map above coincides with the oscillatory integral on G ⋉ R G , as given inDefinition 1.31. This is our Fubini-type result in the context of semi-direct productof tempered pairs: Proposition . Within the context of Lemma 1.38, assume further thatthe groups G and G come from admissible and tame tempered pairs ( G , S ) and ( G , S ) and, with S defined in (1.29) , that (cid:0) G ⋉ R G , S (cid:1) is admissible, tame andtempered too. Assume last that the extension homomorphism R ∈ Hom (cid:0) G , Aut ( G ) (cid:1) , preserves the Haar measure d G and be such that the map [ g
7→ | R g | ] is temperedon ( G , S ) . Let also µ, µ be tempered weights on the semi-direct product G ⋉ R G . Then, for F ∈ B µ ( G ⋉ R G , E ) , m ∈ B µ ( G ⋉ R G ) , with ˆ F and ˆ m the associatedfunctions on G × G as in (1.27) , we have (1.30) ^ Z G ⋉ R G Em (cid:0) F (cid:1) = ^ Z G E (cid:16) ^ Z G E (cid:0) ˆ m ˆ F (cid:1)(cid:17) . Proof.
Since the map [ g
7→ | R g | ] is tempered on ( G , S ), by Lemma 1.5and Lemma 1.21 one deduces that d , the restriction of the modular weight on G ⋉ R G to G is tempered on ( G , S ). Thus by Lemma 1.38, the right hand-side of (1.30) is well defined as a continuous linear map from B µ.µ φ ( G ⋉ R G , E )to E ( µ φ is the tempered weight on G ⋉ R G associated with tameness). Notealso that, by our assumptions, the pair ( G ⋉ R G , S ) is tempered and admissible,the left hand side of (1.30) is also well defined as a continuous linear map from B µ.µ φ ( G ⋉ R G , E ) to E , too. Now, take F ∈ D ( G ⋉ R G , E ) and associate to itˆ F ∈ D (cid:0) G , D ( G , E ) (cid:1) as in (1.27). By construction, we have ^ Z G ⋉ R G Em ( F ) = Z G ⋉ R G E ( g ) m ( g ) F ( g ) d G ⋉ R G ( g ) . Since the extension homomorphism R preserves d G , we have for g ∈ G , g ∈ G :d G ⋉ R G ( g g ) = d G ( g ) d G ( g ) , which, by the ordinary Fubini Theorem, implies that ^ Z G ⋉ R G Em ( F ) = Z G E ( g ) (cid:16) Z G E ( g ) m ( g g ) F ( g g ) d G ( g ) (cid:17) d G ( g )= ^ Z G E (cid:16) ^ Z G E ( ˆ m ˆ F ) (cid:17) . Thus, both sides of (1.30) are continuous linear maps from B µ.µ φ ( G ⋉ R G , E ) to E and coincide on D ( G ⋉ R G , E ). Therefore, these maps coincide on the closureof D ( G ⋉ R G , E ) inside B µ.µ φ ( G ⋉ R G , E ). One concludes using Lemma 1.12(vi), which shows that the latter closure contains B µ ( G ⋉ R G , E ). (cid:3) In this section, we introduce a Schwartz type functions space, out of an admis-sible tempered pair (
G, S ) and prove that it is Fr´echet and nuclear. Our notionof Schwartz space is of course closely related, if not in many cases equivalent, toother notions of Schwartz space on Lie groups, but the point here is that it is for-mulated in terms of the phase function S only. This is this formulation that allowsto immediately implement the compatibility with our notion of oscillatory integral. Definition . Let (
G, S ) be a tempered pair. For all X ∈ U ( g ), we let α X := E − e X E ∈ C ∞ ( G ), where E is defined in (1.17). Then we set S S ( G ) := (cid:8) f ∈ C ∞ ( G ) : ∀ X, Y ∈ U ( g ) , ∀ n ∈ N , sup x ∈ G (cid:12)(cid:12) α nX ( x ) (cid:0) e Y f (cid:1) ( x ) (cid:12)(cid:12) < ∞ (cid:9) . We first prove that this space is isomorphic to the ordinary Schwartz space ofthe Euclidean space g ⋆ . .5. A SCHWARTZ SPACE FOR TEMPERED PAIRS 35 Lemma . Let φ : G → g ⋆ be the diffeomorphism underlying Definition 1.22,associated to an admissible tempered pair ( G, S ) . Fixing an Euclidean structure on g ⋆ , denote by S ( g ⋆ ) the ordinary Schwartz space of g ⋆ . Then, S S ( G ) coincides with S φ ( G ) := (cid:8) f ∈ C ∞ ( G ) : f ◦ φ − ∈ S ( g ⋆ ) (cid:9) . In particular, endowed with the transported topology, S S ( G ) is a nuclear Fr´echetspace. If moreover the pair ( G, S ) is tame, then the transported topology is equiva-lent to the one associated with the semi-norms k . k k,n : f ∈ S S ( G, E ) sup X ∈ U k ( g ) sup x ∈ G n µ φ ( x ) n (cid:12)(cid:12) e X f ( x ) (cid:12)(cid:12) | X | k o , k, n ∈ N , Proof.
Recall that f ∈ S φ ( G ) if and only if for all α, β ∈ N dim( G ) , we have(1.31) sup ξ ∈ g ⋆ (cid:12)(cid:12) ξ α ∂ β ( f ◦ φ − )( ξ ) (cid:12)(cid:12) < ∞ , while f ∈ S S ( G ) if and only if for all X, Y ∈ U ( g ) and all n ∈ N (1.32) sup x ∈ G (cid:12)(cid:12) α nX ( x ) (cid:0) e Y f (cid:1) ( x ) (cid:12)(cid:12) < ∞ . Fix { X j } dim( G ) j =1 a basis of g and let { ξ j } dim( G ) j =1 be the dual basis on g ⋆ . From thesame methods as in Lemma 1.21, one can construct an invertible matrix M ( ξ ) whichis tempered with tempered inverse and which is such that in the φ -coordinates e X j = dim( G ) X i =1 M ( ξ ) j,i ∂ ξ i . Since by Remark 1.23 S is tempered, for all X ∈ U ( g ), the associated multiplier α X = E − e X E in φ -coordinates is bounded by a polynomial function on g ⋆ . Last,since the pair ( G, S ) is admissible, associated to the vector space decomposition g = L Nk =0 V k , there exist elements X k ∈ U ( V k ) and constants C k , ρ k > α k = E − f X k E and with | . | k the Euclidean norm on V k , we have | x n | n ≤ (cid:0) C − n | α n | − (cid:1) /ρ n . Summing up over k = 1 , . . . , N gives the existence of C, ρ >
0, such that | ξ | ≤ C (cid:16) N X k =0 (cid:12)(cid:12) α k (cid:0) φ − ( ξ ) (cid:1)(cid:12)(cid:12)(cid:17) ρ . Putting these three facts together gives the equality between the two sets of func-tions on G and the equivalence of the topologies associated with the semi-norms(1.31) and (1.32).Last, assume that the pair ( G, S ) is also tame, with associated weight µ φ . Thenthere exist constants C , C , ρ , ρ such that, with φ : G → g ⋆ the diffeomorphismunderlying Definition 1.22, we have C (1 + | φ | ) ρ / ≤ µ φ ≤ C (1 + | φ | ) ρ / . which is enough to prove the last claim. (cid:3) More generally, when E is a complex Fr´echet space with topology underlyinga countable set of semi-norms {k . k j } j ∈ N , we define the E -valued Schwartz spaceassociated to an admissible tempered and tame pair ( G, S ) as the Fr´echet spacecompletion of D ( G, E ) for the topology underlying the family of semi-norms: k f k j,k,n := sup X ∈ U k ( g ) sup x ∈ G n µ φ ( x ) n (cid:13)(cid:13) e X f ( x ) (cid:13)(cid:13) j | X | k o , j, k, n ∈ N , Note that by nuclearity of S S ( G ), we have S S ( G, E ) = S S ( G ) ˆ ⊗E (for any completedtensor product). We now present several results which establish most of the analytical propertieswe will need to construct our universal deformation formula for actions of K¨ahleriangroups on Fr´echet algebras. In all that follows, when considering a Fr´echet algebra A with topology underlying a countable set of semi-norms {k . k j } j ∈ N , we will alwaysassume the latter to be sub-multiplicative, i.e . k ab k j ≤ k a k j k b k j , ∀ a, b ∈ A , ∀ j ∈ N . We start with a crucial fact. Its proof being very similar to those of Lemma 1.15,we omit it.
Lemma . Let A be a Fr´echet algebra and let µ , µ be two families of weightson G with sub-multiplicative degree respectively denoted by ( L , R ) and ( L , R ) .Then the bilinear mapping R ⊗ R : C ∞ ( G, A ) × C ∞ ( G, A ) → C ∞ (cid:0) G × G, C ∞ ( G, A ) (cid:1) , ( F , F ) h ( x, y ) ∈ G × G ( R ⋆x F )( R ⋆y F ) := (cid:2) g ∈ G F ( gx ) F ( gy ) (cid:3) ∈ A i , is continuous from B µ ( G, A ) × B µ ( G, A ) to B ν (cid:0) G × G , B λ ( G, A ) (cid:1) , where thefamilies ν := { ν j,k } j,k ∈ N and λ := { λ j } j ∈ N are given by ν j,k := (cid:0) µ ,j ⊗ µ ,j (cid:1) max( R ,j ,R ,j ) d kG × G , λ j := µ L ,j ,j µ L ,j ,j . More precisely, labeling by ( j, k ) ∈ N the semi-norm k . k j,k,λ of B λ ( G, A ) , for all ( j, k, k ′ ) in N , there exists C > such that for all F ∈ B µ ( G, A ) , F ∈ B µ ( G, A ) ,we have (cid:13)(cid:13) R ⊗ R ( F , F ) (cid:13)(cid:13) ( j,k ) ,k ′ ,ν ≤ C k F k j,k + k ′ ,µ k F k j,k + k ′ ,µ . Theorem . Let ( G × G, S ) be an admissible and tame tempered pair. Letalso m ∈ B µ ( G × G ) for some tempered weight µ on G × G and let µ , µ betwo families of weights on G with sub-multiplicative degree respectively denoted by ( L , R ) and ( L , R ) , such that the family of weights µ ⊗ µ is tempered on G × G .Then, for any Fr´echet algebra A , the oscillatory integral ⋆ S := h ( F , F ) ^ Z G × G mE ◦ R ⊗ R ( F , F ) i , (1.33) defines a continuous bilinear map from B µ ( G, A ) × B µ ( G, A ) to B λ ( G, A ) , wherethe family λ := { λ j } j ∈ N , is given by λ j := µ L ,j ,j µ L ,j ,j . .6. BILINEAR MAPPINGS FROM THE OSCILLATORY INTEGRAL 37 More precisely, for any ( j, k ) ∈ N there exist C > and l ∈ N such that for any F ∈ B µ ( G, A ) and F ∈ B µ ( G, A ) , we have k F ⋆ S F k j,k,λ ≤ C k F k j,l,µ k F k j,l,µ . In particular, one has a continuous bilinear product (not necessarily associative!): ⋆ S : B ( G, A ) × B ( G, A ) → B ( G, A ) . Proof.
By Lemma 1.42, the map
R ⊗ R : B µ ( G, A ) × B µ ( G, A ) → B ν (cid:0) G × G , B λ ( G, A ) (cid:1) , where ν j,k = (cid:0) µ ,j ⊗ µ ,j (cid:1) max( R ,j ,R ,j ) d kG × G and λ j = µ L ,j ,j µ L ,j ,j , is a continuousbilinear mapping. By tameness, the family of tempered weights µ.ν is dominated.Hence the oscillatory integral composed with the map R ⊗ R is well defined as acontinuous bilinear mapping from B µ ( G, A ) × B µ ( G, A ) to B λ ( G, A ). The pre-cise estimate follows by putting together Proposition 1.29, Lemma 1.42 and thecontinuity relation k . k j,k, ˆ µ ≤ C k . k j,k,µ , for any family of weights ˆ µ that dominatesanother family µ . (cid:3) We now discuss some issues regarding associativity of the bilinear mapping ⋆ S .To this aim, we need to show how to compute the product F ⋆ S F as the limit ofa double sequence of products of smooth compactly supported functions. Lemma . Within the context of Theorem 1.43, for F ∈ B µ ( G, A ) and F ∈ B µ ( G, A ) , we let { F ,n } , { F ,n } be two sequences in D ( G, A ) convergingrespectively to F and F for the topologies of B ˆ µ ( G, A ) and B ˆ µ ( G, A ) where ˆ µ ≻ µ , ˆ µ ≻ µ are such that ˆ µ ⊗ ˆ µ is tempered on G × G . Setting then λ j := µ L ,j ,j µ L ,j ,j , we have the equalities in B λ ( G, A ) : F ⋆ S F = lim n →∞ lim n →∞ F ,n ⋆ S F ,n = lim n →∞ lim n →∞ F ,n ⋆ S F ,n . Proof.
Note that the family ˆ ν := { (ˆ µ ,j ⊗ ˆ µ ,j ) max( R ,j ,R ,j ) d kG × G } ( j,k ) istempered (by assumption) and dominates the (tempered by assumption) family ν := { ( µ ,j ⊗ µ ,j ) max( R ,j ,R ,j ) d kG × G } ( j,k ) and consequently (see Remark 1.13),we may view R ⊗ R ( F , F ) as an element of B ˆ ν (cid:0) G × G , B λ ( G, A ) (cid:1) , with λ := { µ L ,j ,j µ L ,j ,j } j ∈ N . By the estimate of Theorem 1.43, we know that for all j, k ∈ N ,there exists l ∈ N such that k F ⋆ S F k j,k,λ ≤ C ( k, j ) k F k j,l, ˆ µ k F k j,l, ˆ µ . We then write k F ⋆ S F − F ,n ⋆ S F ,n k j,k,λ ≤ k ( F − F ,n ) ⋆ S F k j,k,λ + k F ,n ⋆ S ( F − F ,n ) k j,k,λ ≤ C ( k, j ) (cid:0) k F − F ,n k j,l, ˆ µ k F k j,l, ˆ µ + k F ,n k j,l, ˆ µ k F − F ,n k j,l, ˆ µ (cid:1) ≤ C ′ ( k, j ) (cid:0) k F − F ,n k j,l, ˆ µ k F k j,l,µ + k F ,n k j,l, ˆ µ k F − F ,n k j,l, ˆ µ (cid:1) , where in the last inequality we used Remark 1.13 which shows that the semi-norm k . k j,k, ˆ µ is dominated by k . k j,k,µ . The latter remark also shows that the numericalsequence {k F ,n k j,l, ˆ µ } n ∈ N is bounded since k F ,n k j,l, ˆ µ ≤ k F ,n − F k j,l, ˆ µ + k F k j,l, ˆ µ ≤ k F ,n − F k j,l, ˆ µ + C k F k j,l,µ . This completes the proof. (cid:3)
Remark . In other words, within the setting of Lemma 1.44, we have in B λ ( G, A ): F ⋆ S F ′ = lim m,n →∞ Z G × G E ( x, x ′ ) m ( x, x ′ ) R ⋆x ( F m ) R ⋆x ′ ( F ′ n ) d G ( x ) d G ( x ′ ) , for suitable approximation sequences { F n } , { F ′ n } ⊂ D ( G, A ). Definition . Within the context of Theorem 1.43, we say that the product ⋆ S , given in (1.33), is weakly associative when for all ψ , ψ , ψ ∈ D ( G, A ), onehas ( ψ ⋆ S ψ ) ⋆ S ψ = ψ ⋆ S ( ψ ⋆ S ψ ) in B ( G, A ). Proposition . Within the context of Theorem 1.43, weak associativityimplies strong associativity in the sense that, when weakly associative, for everyfurther family of weights µ on G with sub-multiplicative degree denoted by ( L , R ) such that µ ⊗ µ is tempered on G × G . Then, for every element ( F , F , F ) ∈B µ ( G, A ) × B µ ( G, A ) × B µ ( G, A ) , one has the equality ( F ⋆ S F ) ⋆ S F = F ⋆ S ( F ⋆ S F ) in B λ ( G, A ) for λ = { µ L ,j ,j µ ,jL ,j µ ,jL ,j } j ∈ N . Proof.
Let µ φ be a tempered weight on G × G which dominates the constantweight 1 (it exists by assumption of tameness.) Consider the element ν φ ∈ C ∞ ( G )defined by ν φ ( g ) := µ φ ( g, e ). The latter is then (by a direct application of Lemma1.5) a weight on G that dominates 1. Moreover, it is easy to see that ν φ ⊗ ν φ istempered on G × G . Hence, all the family of weights µ , µ and µ are dom-inated e.g . by ˆ µ := ν φ .µ , ˆ µ := ν φ .µ and ˆ µ := ν φ .µ and ˆ µ ⊗ ˆ µ andˆ µ ⊗ ˆ µ are tempered on G × G . The assumptions of Lemma 1.44 are there-fore satisfied. Let us consider sequences of smooth compactly supported elements { Φ ,n } n ∈ N , { Φ ,n } n ∈ N and { Φ ,n } n ∈ N that converge to the elements F , F and F respectively in B ˆ µ ( G, A ) , B ˆ µ ( G, A ) and B ˆ µ ( G, A ). Using separate continuity of ⋆ S and Lemma 1.44, we observe the following equality:lim n →∞ (cid:16) lim n →∞ (cid:16) lim n →∞ (Φ ,n ⋆ S Φ ,n ) ⋆ S Φ ,n (cid:17)(cid:17) = ( F ⋆ S F ) ⋆ S F , in B λ ( G, A ) for λ = { µ L ,j ,j µ ,jL ,j µ ,jL ,j } . One then concludes using weak asso-ciativity and the commutativity of the limits, as shown in Lemma 1.44. (cid:3) In section 1.5, we have seen how to associate in a canonical way a Schwartztype functions space to a tempered, admissible and tame pair. Hence, starting withsuch a pair ( G × G, S ), we get a Schwartz space on G × G . But we can also definea one-variable Schwartz space using the continuity of the partial evaluation maps: Definition . Let ( G × G, S ) be a tempered admissible and tame pair and A be a Fr´echet algebra. We define the A -valued Schwartz space on G associated to S by S S ( G, A ) := n(cid:2) g ∈ G f ( g, e ) (cid:3) , f ∈ S S ( G × G, A ) o . We endow the latter with the topology induced by the semi-norms: k f k j,k,n := sup X ∈ U k ( g ) sup x ∈ G n µ φ, ( x ) n (cid:13)(cid:13) e X f ( x ) (cid:13)(cid:13) j | X | k o , j, k, n ∈ N , (1.34) .6. BILINEAR MAPPINGS FROM THE OSCILLATORY INTEGRAL 39 with µ φ, ( x ) := µ φ ( x, e ) and µ φ is the tempered weight on G × G associated withthe tameness (Definition 1.34).The next Lemma shows that the right action on the space of A -valued Schwartzfunctions, leads us to a B -type space for family of weights too. Lemma . Let ( G × G, S ) be a tame and admissible tempered pair, A be aFr´echet algebra and µ be a family of weights on G with sub-multiplicative degree ( L, R ) , which is bounded by a power of µ φ, (see Definition 1.48). Then, thereexists a sequence { M n } n ∈ N of integers, such that for all elements F ∈ B µ ( G, A ) and ϕ ∈ S S ( G, A ) , the element ( R ⊗ R )( F, ϕ ) (defined in Lemma 1.42) belongs to B ν ( G × G, S S ( G, A )) where ν := { ν j,k,n } j,k,n ∈ N with ν j,k,n := (cid:0) µ R j j ⊗ µ ∨ M n φ, (cid:1) d kG × G , j, k, n ∈ N . Proof.
Denote by k . k j,k,n , ( j, k, n ) ∈ N , the semi-norms (1.34) of S S ( G, A ).Then, the semi-norms of an elementΦ = (cid:2) ( x, y ) ∈ G × G [ g ∈ G Φ( x, y ; g ) ∈ A ] (cid:3) ∈ B ν (cid:0) G × G, S S ( G, A ) (cid:1) , are given by k Φ k ( j,k,n ) , ( k ,k ) ,ν =sup x,y,g ∈ G sup X ∈ U k ( g ) sup ( Y ,Y ) ∈ U k ( g ) ×U k ( g ) µ φ, ( g ) n k e X g ( e Y ⊗ e Y ) ( x,y ) Φ( x, y ; g ) k j ν j,k,n ( x, y ) . Using Sweedler’s notation (0.2), we have for X ∈ U k ( g ), Y ∈ U k ( g ), Y ∈ U k ( g ): e X g . (cid:16) ( e Y ⊗ e Y ) ( x,y ) (cid:0) R ⋆x F ( g ) R ⋆y ϕ ( g ) (cid:1)(cid:17) = X ( X ) (cid:0) ^ ( Ad x − X ) e Y F (cid:1) ( gx ) (cid:0) ^ ( Ad y − X ) e Y ϕ (cid:1) ( gy ) , which yields the following estimation for arbitrary N ∈ N : k e X g . (cid:16) ( e Y ⊗ e Y ) ( x,y ) . (cid:0) R ⋆x F ( g ) R ⋆y ϕ ( g ) (cid:1)(cid:17) k j ≤ X ( X ) | X (1) | k | X (2) | k | Ad x − | k | Ad y − | k | Y | k | Y | k × sup Z ∈ U k + k ( g ) k e Z F ( gx ) k j | Z | k + k sup Z ∈ U k + k ( g ) k e Z ϕ ( gy ) k j | Z | k + k ≤ X ( X ) | X (1) | k | X (2) | k | Ad x − | k | Ad y − | k | Y | k | Y | k × µ j ( gx ) µ − Nφ, ( gy ) k F k j,k + k ,µ k ϕ k j,k + k ,N , which by Lemma 1.5, Lemma 1.14 and the estimate (1.6) is bounded by a constanttimes(1.35) | X | k | Y | k | Y | k d G × G ( x, y ) k µ j ( gx ) µ − Nφ, ( gy ) k F k j,k + k ,µ k ϕ k j,k + k ,N . Setting (
L, R ) for the sub-multiplicative degree of µ φ and using µ − φ, ( gy ) ≤ C /L µ − /Lφ, ( g ) µ R/Lφ, ( y − ) , y, g ∈ G , we see that (1.35) is (up to a constant) bounded by | X | k | Y | k | Y | k d G × G ( x, y ) k µ L j j ( g ) µ R j j ( x ) µ − N/Lφ, ( g ) µ NR/Lφ, ( y − ) × k F k j,k + k ,µ k ϕ k j,k + k ,N . Now, given n ∈ N , chose N n , M n ∈ N such that µ L j j µ − N n /Lφ, ≤ µ − nφ, , and µ N n R/Lφ, ≤ µ M n φ, . Then, set ν j,k,n ( x, y ) := µ R j j ( x ) µ ∨ M n φ, ( y ) d kG × G ( x, y ) , j, k, n ∈ N . This entails that (1.35) is bounded by | X | k | Y | k | Y | k µ − nφ, ( g ) µ j,k,n ( x, y ) k F k j,k + k ,µ k ϕ k j,k + k ,N n , which, for a finite constant C ( j, k, n, k , k ) >
0, finally gives k ( R ⊗ R )( F, ϕ ) k ( j,k,n ) , ( k ,k ) ,ν ≤ C ( j, k, n, k , k ) k F k j,k + k ,µ k ϕ k j,k + k ,N n , proving the claim. (cid:3) Observe that 1 ⊗ µ φ, is tempered on G × G . By Remark 1.2 and since theinversion map on a tempered group is a tempered map, we deduce that 1 ⊗ µ ∨ φ, is tempered on G × G too. Hence, when µ ⊗ G × G , so isthe family of weights ν given in the Lemma 1.49. We then deduce the followingimportant consequence of the latter: Proposition . Let ( G × G, S ) be a tame and admissible tempered pair, A a Fr´echet algebra, µ a family of tempered weights on G , such that the family ofweights µ ⊗ is tempered on G × G and m ∈ B µ ( G × G ) for some tempered weight µ on G × G . Then the bilinear map ⋆ S , defined in (1.33) , is continuous on S S ( G, A ) and one has the continuous bilinear map: ⋆ S : B µ ( G, A ) × S S ( G, A ) → S S ( G, A ) , ( F, ϕ ) L ⋆ S ( F ) : ϕ F ⋆ S ϕ . Remark . In the context of the proposition above, observe that the re-striction to S S ( G, A ) × S S ( G, A ) of the bilinear product ⋆ S (1.33), we have the(point-wise and semi-norm-wise) absolutely convergent expression: ϕ ⋆ S ϕ = Z G × G m ( x , x ) E ( x , x ) R ⋆x ′ ( ϕ ) R ⋆x ′ ( ϕ ) d G ( x ) d G ( x ) . HAPTER 2
Tempered pairs for K¨ahlerian Lie groups
The aim of this chapter is to endow each negatively curved K¨ahlerian Lie groupwith the structure of a tempered, tame and admissible pair. Recall that a Lie group G is called a K¨ahlerian Lie group when it is endowed with an invariant K¨ahlerstructure, i.e . a left-invariant complex structure J together with a left-invariantRiemannian metric g such that the triple ( G, J , g ) constitutes a K¨ahler manifold.Within the present memoir, we will be concerned with K¨ahlerian Lie groups whosesectional curvature is negative. We call them negatively curved .In section 2.1, we briefly review the theory of normal j -algebras and associatednormal j -groups, which in turn gives a classification result for negatively curvedK¨ahlerian Lie groups. In section 2.2, we explain how an elementary normal j -group is naturally endowed with a structure of a symplectic symmetric space. Suchstructure is the core of the construction of chapter 5. It is also in this context thatwe have a clear geometric construction for the phase and amplitude of the kernelunderlying our deformation formula. It is then in sections 2.3 and 2.4 that we provethat the phase mentioned above endows a negatively curved K¨ahlerian Lie groupwith the structure of an admissible and tempered pair. The following definition, due to Pyatetskii-Shapiro [ ], describes the infinites-imal structure of negatively curved K¨ahlerian Lie groups. Definition . A normal j-algebra is a triple ( b , α, j ) where(i) b is a solvable Lie algebra which is split over the reals, i.e . ad X has onlyreal eigenvalues for all X ∈ b ,(ii) j is an endomorphism of b such that j = − X, Y ] + j [ j X, Y ] + j [ X, j Y ] − [ j X, j Y ] = 0 , X, Y ∈ b , (iii) α is a linear form on b such that α ([ j X, X ]) > X = 0 and α ([ j X, j Y ]) = α ([ X, Y ]) , X, Y ∈ b . We quote the following structure result from [ ]. Proposition . The Lie algebra of a negatively curved K¨ahlerian Lie groupalways carries a structure of normal j -algebra. If b ′ is a subalgebra of b which is invariant by j , then ( b ′ , α | b ′ , j | b ′ ) is again a nor-mal j -algebra, called a j-subalgebra of ( b , α, j ). A j -subalgebra whose underlyingLie algebra b ′ is an ideal of b is called a j -ideal. Example . Every Iwasawa factor AN of the simple Lie group SU (1 , n ) isnaturally a negatively curved K¨ahlerian Lie group. Indeed, denoting by K ≃ U ( n )
412 2. TEMPERED PAIRS FOR K¨AHLERIAN LIE GROUPS a maximal compact subgroup of SU (1 , n ), one knows that the associated symmet-ric space G/K is a negatively curved K¨ahlerian SU (1 , n )-manifold. The associatedIwasawa decomposition SU (1 , n ) = AN K then yields a global diffeomorphism be-tween
G/K and AN . Transporting to AN the K¨ahler structure of G/K under thelatter diffeomorphism, then endows AN with a negatively curved K¨ahlerian Liegroup structure, called elementary after Pyatetskii-Shapiro.The infinitesimal structure underlying an elementary normal j -group (cf . theabove Example 2.3) may be precisely described as follows. Let ( V, ω ) be a sym-plectic vector space of real dimension 2 d . We consider the associated HeisenbergLie algebra h := V ⊕ R E . That is, h is the central extension of the Abelian Liealgebra V , with brackets given by[ v , v ] := ω ( v , v ) E , v , v ∈ V , [ E, X ] := 0 , X ∈ h . Definition . Let a be a one-dimensional real Lie algebra, with generator H . We consider the split extension of Lie algebras:0 → h → s := a ⋉ ρ h h → a → , with extension homomorphism ρ h : a → Der( h ) given by(2.1) ρ h ( H ) (cid:0) v + t E (cid:1) := [ H, v + t E ] := v + 2 t E , v ∈ V , t ∈ R . The Lie algebra s is called elementary normal . Last, we denote by S the con-nected simply connected Lie group whose Lie algebra is s and we call the latter an elementary normal j-group .Note that S is a solvable group of real dimension 2 d + 2 and if V = { } , S isisomorphic to the affine group of the real line. It turns out that every negativelycurved K¨ahlerian Lie group can be decomposed into elementary pieces: at theinfinitesimal level, one has the following result, due to Pyatetskii-Shapiro [ ]. Proposition . Let ( b , α, j ) be a normal j -algebra. Then, there exist z , aone-dimensional ideal of b and V , a vector subspace of b , such that setting a := j z ,the algebra s := a ⊕ V ⊕ z underlies an elementary normal j -ideal of b . Moreover,the associated extension sequence −→ s −→ b −→ b ′ −→ , is split as a sequence of normal j -algebras and such that: (2.2) [ b ′ , a ⊕ z ] = 0 and [ b ′ , V ] ⊂ V .
In particular, every normal j -algebra b admits a decomposition as a sequence ofsplit extensions of elementary normal j -algebras s i , i = 1 , . . . , N , of real dimension2 d i + 2, d i ∈ N : (cid:0) . . . (cid:0) s N ⋉ s N − (cid:1) ⋉ · · · ⋉ s (cid:1) ⋉ s , such that for all i = 1 , . . . , N − (cid:2)(cid:0) s N ⋉ . . . (cid:1) ⋉ s i +1 , a i ⊕ z i (cid:3) = 0 and (cid:2)(cid:0) s N ⋉ . . . (cid:1) ⋉ s i +1 , V i (cid:3) ⊂ V i . Definition . A normal j-group B , consists of a connected simply con-nected Lie group that admits a normal j -algebra as Lie algebra, i.e . B = exp { b } ,where b is a normal j -algebra. .2. GEOMETRIC STRUCTURES ON ELEMENTARY NORMAL j -GROUPS 43 At the group level, for i = 1 , . . . , N −
1, call R i the extension homomorphismat each step: R i ∈ Hom (cid:0) ( S N ⋉ . . . ) ⋉ S i +1 , Aut ( S i ) (cid:1) . (2.3)The conditions given in (2.2) implies that R i takes values in Sp( V i , ω i ), where( V i , ω i ) denotes the symplectic vector space attached to S i . In this section, we review the properties of a symplectic symmetric space struc-ture every elementary normal j -group is naturally endowed with. The phase func-tion with respect to which an admissible tempered pair will be associated to lateron, was defined in [ ] in terms of this symplectic symmetric space structure. Westart with the definition of a symplectic symmetric space as in [ ] which is an adap-tation to the symplectic case of the notion of symmetric space as introduced by O.Loos [ ]. Definition . A symplectic symmetric space is a triple ( M, s, ω ) where(i) M is a connected smooth manifold,(ii) s is a smooth map s : M × M → M , ( x, y ) s x ( y ) := s ( x, y ) , such that:(ii.1) For every x ∈ M , the partial map s x : M → M is an involutivediffeomorphism admitting x as isolated fixed point. The diffeomorphism s x is called the symmetry at point x .(ii.2) For all points x and y in M , the following relation holds: s x ◦ s y ◦ s x = s s x ( y ) . (iii) ω is a closed and non-degenerate two-form on M that is invariant underthe symmetries: s ⋆x ω = ω , ∀ x ∈ M . A morphism between two symplectic symmetric spaces is defined as a symplecto-morphism that intertwines the symmetries.Symplectic symmetric spaces always carry a preferred Lie group of transforma-tions [ ]: Definition . The automorphism group of a symplectic symmetric space(
M, s, ω ) is constituted by the symplectomorphisms ϕ ∈ Symp(
M, ω ) which arecovariant under the symmetries: ϕ ◦ s x = s ϕ ( x ) ◦ ϕ, ∀ x ∈ M .
It is a Lie subgroup of Symp(
M, ω ) that acts transitively on M and it is denotedby Aut ( M, s, ω ). Its Lie algebra is called the derivation algebra of (
M, s, ω ) andis denoted by aut ( M, s, ω ). The closedness condition is, in fact, redundant, see [ ]. We now pass to the particular case of a given 2 d + 2-dimensional elementarynormal j -group S with associated symplectic form ω S . Let a, t ∈ R and v ∈ V ≃ R d .The following identification will always be understood: R d +2 → s , x := ( a, v, t ) aH + v + tE . The following result is extracted from [ , , ]: Proposition . Let S be an elementary normal j -group. (i) The map (2.4) s → S , ( a, v, t ) exp( aH ) exp( v + tE ) = exp( aH ) exp( v ) exp( tE ) , is a global Darboux chart on ( S , ω S ) in which the symplectic structurereads: ω S := 2d a ∧ d t + ω . (ii) Setting furthermore s ( a,v,t ) ( a ′ , v ′ , t ′ ) :=(2.5) (cid:0) a − a ′ , v cosh( a − a ′ ) − v ′ , t cosh(2 a − a ′ ) − t ′ + ω ( v, v ′ ) sinh( a − a ′ ) (cid:1) , defines a symplectic symmetric space structure ( S , s, ω S ) on the elementarynormal j -group S . (iii) The left action L x : S → S , x ′ x.x ′ , defines a injective Lie grouphomomorphism L : S → Aut ( S , s, ω S ) . In the coordinates (2.4) , we have x.x ′ = ( a, v, t ) . ( a ′ , v ′ , t ′ ) = (cid:0) a + a ′ , e − a ′ v + v ′ , e − a ′ t + t ′ + e − a ′ ω ( v, v ′ ) (cid:1) . and x − = ( a, v, t ) − = ( − a, − e a v, − e a t ) . (iv) The action R : Sp( V, ω ) × S → S , ( A, ( a, v, t )) R A ( a, v, t ) := ( a, Av, t ) by automorphisms of the normal j -group S induces an injective Lie grouphomomorphism: R : Sp( V, ω ) → Aut ( S , s, ω S ) , A R A . In fact,
Sp(
V, ω ) ≃ Aut ( S ) ∩ Aut ( S , s, ω S ) . Note that in the coordinates (2.4), the modular function of S , ∆ S , reads e (2 d +2) a .We now pass to the definition of the three-point phase on S . For this we needthe notion of “double geodesic triangle” as introduced by A. Weinstein [ ] and Z.Qian [ ]. Definition . Let (
M, s ) be a symmetric space. A midpoint map on M is a smooth map M × M → M , ( x, y ) mid ( x, y ) , such that, for all points x, y in M : s mid ( x,y ) ( x ) = y . Remark . When it exists, such a midpoint map on a symmetric space(
M, s ) is necessarily unique (see Lemma 2.1.6 of [ ]). .2. GEOMETRIC STRUCTURES ON ELEMENTARY NORMAL j -GROUPS 45 Remark . Observe that in the case where the partial maps s y : M → M , x s x ( y ) are global diffeomorphisms of M , a midpoint map exists and is given by: mid ( x, y ) := ( s x ) − ( y ) . Note that in this case, every ϕ ∈ Aut ( M, s ) intertwines the midpoints. Indeed,since for all x, y ∈ M we have ϕ ( s y ( x )) = s ϕ ( y ) (cid:0) ϕ ( x ) (cid:1) , we get ϕ (cid:0) mid ( x, y ) (cid:1) = mid (cid:0) ϕ ( x ) , ϕ ( y ) (cid:1) . An immediate computation shows that a midpoint map always exists on thesymplectic symmetric space attached to an elementary normal j -group: Lemma . For the symmetric space ( S , s ) underlying an elementary normal j -group, the associated partial maps are global diffeomorphisms. In the coordinates (2.4) , we have: (cid:0) s ( a ,v ,t ) (cid:1) − ( a, v, t ) = (cid:16) a + a , v + v a − a ) , t + t a − a ) − ω ( v, v ) sinh( a − a a − a ) cosh( a − a (cid:17) . The following statement is proved in [ ]. Proposition . Let S be an elementary normal j -group. (i) The K¨ahler manifold S is strictly geodesically complete: two points deter-mine a unique geodesic arc. (ii) The “medial triangle” three-point function
Φ : S → S , ( x , x , x ) (cid:0) mid ( x , x ) , mid ( x , x ) , mid ( x , x ) (cid:1) , is a S -equivariant (under the left regular action) global diffeomorphism. Since our space S has trivial de Rham cohomology in degree two, any threepoints ( x, y, z ) ∈ S define an oriented geodesic triangle T ( x, y, z ) whose symplec-tic area is well-defined by integrating the two-form ω S on any surface admitting T ( x, y, z ) as boundary. With a slight abuse of notation, we setArea( x, y, z ) := Z T ( x,y,z ) ω S . Definition . The canonical two-point phase associated to an elemen-tary normal j -group is defined by S S can ( x , x ) := Area (cid:0) Φ − ( e, x , x ) (cid:1) ∈ C ∞ ( S , R ) , where e := (0 , ,
0) denotes the unit element in S . In the coordinates (2.4), one hasthe explicit expression:(2.6) S S can ( x , x ) = t sinh 2 a − t sinh 2 a + ω ( v , v ) cosh a cosh a . The canonical two-point amplitude associated to an elementary normal j -groupis defined by A S can ( x , x ) := Jac Φ − ( e, x , x ) / ∈ C ∞ ( S , R ) . In the coordinates (2.4), it reads A S can ( x , x ) =(2.7) (cid:0) cosh a cosh a cosh( a − a ) (cid:1) d (cid:0) cosh 2 a cosh 2 a cosh(2 a − a ) (cid:1) / . The aim of this technical section is to prove that the pair ( S × S , S S can ) istempered, admissible and tame. We start by splitting the 2 d -dimensional symplecticvector space ( V, ω ) associated to an elementary normal j -group S into a direct sumof two Lagrangian subspaces in symplectic duality: V = l ⋆ ⊕ l . The following result establishes temperedness.
Lemma . The pair ( S × S , S S can ) is tempered. Moreover, the Jacobian of themap φ : S × S → ( s ⊕ s ) ⋆ , g h X ∈ s ⊕ s (cid:0) e X. S S can (cid:1) ( g ) i , is proportional to ( A S can ) . Proof.
Let us fix { f j } dj =1 , a basis of l ⋆ to which we associate { e j } dj =1 thesymplectic-dual basis of l , i.e . it is defined by ω ( f i , e j ) = δ i,j . We let E the centralelement of the Heisenberg Lie algebra h ⊂ s and H the generator of a in the onedimensional split extension which defines the Lie algebra s :0 → h → s → a → . Accordingly, we consider the following basis of s ⊕ s : H := H ⊕ { } , H := { } ⊕ H ,f j := f j ⊕ { } , f j := { } ⊕ f j ,e j := e j ⊕ { } , e j := { } ⊕ e j ,E := E ⊕ { } , E := { } ⊕ E , where the index j runs from 1 to d = dim( V ) /
2. From Proposition 2.9 iii) and withthe notation v = ( x, y ) ∈ l ⋆ ⊕ l = V , we see that the left-invariant vector fields on S are given by:(2.8) e H = ∂ a − P dj =1 ( x j ∂ x j + y j ∂ y j ) − t∂ t , e f j = ∂ x j − y j ∂ t , e e j = ∂ y j + x j ∂ t , e E = ∂ t . Thus, we find e H S S can = 2 t cosh 2 a + 2 t sinh 2 a − ω ( v , v ) e − a cosh a , (2.9) e H S S can = − t cosh 2 a − t sinh 2 a − ω ( v , v ) e − a cosh a , e E S S can = − sinh 2 a , e E S S can = sinh 2 a , e f j S S can = y j cosh a cosh a + y j sinh 2 a , e f j S S can = − y j cosh a cosh a − y j sinh 2 a , e e j S S can = − x j cosh a cosh a − x j sinh 2 a , e e j S S can = x j cosh a cosh a + x j sinh 2 a . .3. TEMPERED PAIR FOR ELEMENTARY NORMAL j -GROUPS 47 A computation then shows that the Jacobian of the map φ : S × S → ( s ⊕ s ) ⋆ ,underlying Definition 1.22, is given by2 d +2 (cid:0) cosh a cosh a cosh( a − a ) (cid:1) d cosh 2 a cosh 2 a cosh 2( a − a )= 2 d +2 A S can ( x , x ) ≥ d +2 , and hence φ is a global diffeomorphism. It is also clear from Proposition 2.9 iii),that the multiplication and inversion maps on S × S are tempered functions in thecoordinates (2.9). Therefore, the pair ( S × S , S S can ) is tempered. (cid:3) Remark . Note that the formal adjoints of the left invariant vector fields(2.8), with respect to the inner product of L ( S ) read: e H ∗ = − e H + 2 d + 2 , e f ∗ j = − e f j , e e ∗ j = − e e j , e E ∗ = − e E , so that the assumption (1.16) is trivially satisfied.We will now prove that the tempered pair ( S × S , S S can ) is admissible and tame.For this, we need a decomposition of the Lie algebra s and we shall use the followingone:(2.10) s = M k =0 V k where V := a , V := l ⋆ , V := l and V := R E .
Note that both V and V are of dimension one, while V and V are d -dimensional.Accordingly, we consider the decompositions of s ⊕ s given by s ⊕ { } = M k =0 V ,k and { } ⊕ s = M k =0 V ,k , where the subspaces V i,k , i = 1 ,
2, of each factor correspond respectively to thesubspaces V k of s within the decomposition (2.10). We then set:(2.11) V k := V ,k ⊕ V ,k and s ⊕ s = M k =0 V k , by which we mean that there are four subspaces involved in the ordered decompo-sition of s ⊕ s . We also let V ( k ) := k M n =0 V k , k = 0 , , , , as in (1.15) and we let U ( V ( k ) ) be the unital subalgebra of U ( s ⊕ s ) generatedby V ( k ) as in (0.8). Accordingly, we consider the associated tempered coordinates(1.12): x i, := e H i S S can , x ji, := e f ij S S can , x ji, := e e ij S S can , x i, := e E i S S can , with i = 1 , j = 1 , . . . , d and we use the vector notations: ~x := ( x , , x , ) ∈ R , (2.12) ~x := ( x , , x , ) := (cid:0) ( x j , ) dj =1 , ( x j , ) dj =1 ) ∈ R d ,~x := ( x , , x , ) := (cid:0) ( x j , ) dj =1 , ( x j , ) dj =1 ) ∈ R d ,~x := ( x , , x , ) ∈ R . According to the notations ( a, v, t ) ∈ R × R d × R ≃ S and v = ( x, y ) ∈ l ⋆ ⊕ l = V ,we set ~a := ( a , a ) ∈ R , ~x = ( x , x ) ∈ R d , ~y = ( y , y ) ∈ R d , ~t := ( t , t ) ∈ R . We consider the functions s := t sinh 2 a − t sinh 2 a , Ω := ω ( v , v ) , γ := cosh a cosh a , in term of which we have S S can = s + γ Ω . Introducing last A := (cid:18) sinh 2 a cosh 2 a − cosh 2 a − sinh 2 a (cid:19) , (2.13) B := (cid:18) − sinh 2 a − cosh a cosh a cosh a cosh a sinh 2 a (cid:19) ,~γ := (cid:0) e − a cosh a , e − a cosh a (cid:1) , ~δ := (cid:0) − sinh 2 a , sinh 2 a (cid:1) , the relations given in (2.9) can be summarized as:(2.14) ~x = ~δ , ~x = B.~x , ~x = − B.~y , ~x = 2 A.~t − Ω ~γ . We first treat the easiest variable ~x , which lead to multipliers α that satisfyproperty (ii) of Definition 1.24 with constant µ : Lemma . Consider an element X ∈ U ( V ) such that the associated mul-tiplier α X is invertible. Then, for every Y ∈ U ( V (3) ) = U ( s ⊕ s ) there exists apositive constant C Y such that (cid:12)(cid:12) e Y α X (cid:12)(cid:12) ≤ C Y | α X | . Proof.
Note first that V turns out to be a two-dimensional Abelian Liealgebra. Note also that α E i , i = 1 , ~t . Thus, givena two-variables polynomial P , we have for X = P ( E , E ) ∈ U ( V ): α X = P (cid:0) − sinh 2 a , sinh 2 a (cid:1) . It also follows from the explicit expression of the left-invariant vector fields givenin (2.8) that e Y α X = 0 for all Y ∈ U ( ⊕ k =1 V k ). Hence, it suffices to treat the caseof Y ∈ U ( V ). Observe that the restriction of e H j to functions which depend onlyon a j , equals ∂ a j . Thus in this case, we see that e Y α X is a polynomial of the samedegree as P , but in the variables e ± a and e ± a . This is enough to conclude when α X is invertible. (cid:3) Next, we treat the variables ~x and ~x . We first observe Lemma . There exist finitely many matrices B ( r ) ∈ M ( R [ e ± a , e ± a ]) such that for all integers N and N , the elements e H N e H N B consist of a linearcombination of the B ( r ) ’s, where the matrix B has been defined in (2.13) . The sameproperty holds for the matrix A . .3. TEMPERED PAIR FOR ELEMENTARY NORMAL j -GROUPS 49 Proof.
Set D := (cid:18) − sinh 2 a sinh 2 a (cid:19) , Γ := γ (cid:18) −
11 0 (cid:19) , and observes that B = D + Γ and ∂ a i Γ = Γ , i = 1 , . The derivatives of B therefore all belong to the space generated by the entries of D and Γ and by finitely many of their derivatives. This is enough to conclude sincerestricted to functions that depend only on the variable a , we have e H = ∂ a . Theproof for the matrix A is entirely similar. (cid:3) We can now deduce what we need for the variables ~x and ~x . Lemma . There exist finitely many tempered functions m , ( r ) (respectively m , ( r ) ) depending on the variable ~x only, such that for every element X ∈ U ( V (2) ) (respectively X ∈ U ( V (1) ) ), the element e X ~x (respectively e X ~x ) belongs to thespace spanned by { m , ( r ) , m , ( r ) ~x } (respectively { m , ( r ) , m , ( r ) ~x } ). Proof.
This follows from Lemma 2.19 and the expressions (2.8) for the in-variant vector fields. Indeed, the latter implies that for every X ∈ U ( L k =1 V k )(respectively X ∈ U ( V )) of strictly positive homogeneous degree, e X ~x (respec-tively e X ~x ) is either zero or one of the entries of the matrix B . (cid:3) Remark . Note that in view of the expressions (2.8) and (2.9) and bysymmetry on ~x and ~x the assertion in Lemma 2.20 holds for every element X in U ( s ⊕ s ) for both variables ~x and ~x .Last, we go to the variable ~x . The next Lemma is proved using the same typeof arguments as in the proof of Lemma 2.19. Lemma . There exist finitely many vectors γ ( r ) ∈ R [ e ± a , e ± a ] such thatfor all integers N and N , the elements e H N e H N γ consist of a linear combinationof the γ ( r ) ’s. Observing that e H i ~t is proportional to t i and that e H i Ω = − Ω , the Lemmas2.19 and 2.22 then yield the following result. Lemma . There exist finitely many matrices M ( r ) ∈ M ( R [ e ± a , e ± a ]) andfinitely many vectors v ( s ) ∈ R [ e a , a a ] such that for all integers N and N , onehas e H N e H N ~x = M N ,N ~x + Ω v N ,N , with M N ,N ∈ span { M ( r ) } and v N ,N ∈ span { v ( s ) } . The following result is then a consequence of Lemmas 2.18, 2.20 and 2.22.
Corollary . For every k = 0 , . . . , , there exists a tempered function < m k with ∂ ~x j m k = 0 for every j ≤ k and such that for every X ∈ U ( V ( k ) ) ,there exists C X > with (cid:12)(cid:12) e X ~x k (cid:12)(cid:12) ≤ C X m k (1 + | ~x k | ) . Remark . In fact the function m above depends on ~x , ~x , ~x and thefunctions m and m depend on ~x only (and m is constant as it should be). We are now able to check the admissibility conditions of Definition 1.24, forthe tempered pair ( S × S , S S can ). Proposition . Define X := 1 − H − H , X := 1 − d X j =1 (cid:0) ( f j ) + ( f j ) (cid:1) ,X := 1 − d X j =1 (cid:0) ( e j ) + ( e j ) (cid:1) , X := 1 − E − E . Then the corresponding multipliers α k := e − iS S can e X k e iS S can , k = 0 , . . . , , satisfyconditions (i) and (ii) of Definition 1.24. Proof.
We start by observing the following expression of the multiplier: α k = 1 + | ~x k | − iβ k , k = 0 , . . . , , where β k := e X ,k x ,k + e X ,k x ,k , with obvious notations. Then we get1 | α k | = 1(1 + | ~x k | ) + β k ≤ | ~x k | ) , and the first condition of Definition 1.24 is satisfied for C k = 1 and ρ k = 2. Let now X ∈ U ( V ( k ) ) be of strictly positive order. Then, using Sweedler’s (0.2), notationswe get e X α k = X ( X ) (cid:0) e X (1) ~x k (cid:1) . (cid:0) e X (2) ~x k (cid:1) − i e X e X ,k x ,k − i e X e X ,k x ,k . Since X (1) , X (2) , X ,k , X ,k ∈ U ( V ( k ) ), Corollary 2.24 yields | e X α k | ≤ C m k (1 + | ~x k | ) + C m k (1 + | ~x k | ) . As 1 + | ~x k | ≤ | α k | , the second condition of Definition 1.24 is satisfied for µ k = m k (1 + m k ). (cid:3) Last, we prove tameness for the pair ( S × S , S S can ). We start by describing thebehavior of the modular weight of the group S : Lemma . Within the chart (2.4) , we have the following behavior of the themodular weight d S of an elementary normal j -group S : d S ≍ (cid:2) ( a, v, t ) cosh a + cosh 2 a + | v | (1 + e a + cosh a ) + | t | (1 + e a ) (cid:3) . Proof.
Within the decomposition R H ⊕ V ⊕ R E of s , and within the chart(2.4) of S , a quick computation gives Ad ( a,v,t ) = id − e a v e a e a t e a ω ( v, . ) e a , (2.15) Ad ( a,v,t ) − = id v e − a − t − e − a ω ( v, . ) e − a . .4. TEMPERED PAIRS FOR NORMAL j -GROUPS 51 Now, the result follows from the equivalence of the operator and Hilbert-Schmidtnorms on the finite dimensional vector space
End ( s ), together with obvious esti-mates. (cid:3) Corollary . The tempered pair ( S × S , S S can ) is tame. Proof.
Combining the last statement of Lemma 1.5 with Lemma 2.27, we get d S × S ( a , v , t ; a , v , t ) ≥ C (cid:0) cosh a + cosh a + | v | + | v | + | t | + | t | ) , so that with the relations (2.14) in mind, we see that there exists C ′ > (cid:0) | ~x | + | ~x | + | ~x | + | ~x | (cid:1) / ≤ C ′ d S × S . According to Definition 1.34, we may set µ φ = d S × S which is tempered since d S × S is by Lemma 1.5 and Lemma 1.21. (cid:3) We summarize all this by stating the main result of this section:
Theorem . Let S be an elementary normal j -group and let S S can be thesmooth function on S × S given in Definition 2.15. Then, the pair ( S × S , S S can ) istempered, admissible and tame. Remark . From Remark 2.21 and the above discussion, we observe thatsetting V := V ⊕ V yields a decomposition into three subspaces: s ⊕ s = V ⊕ V ⊕ V also underlying admissibility but with associated elements X , X and X := X + X . The corresponding multipliers are α , α and α = α + α . Let B be a normal j -group with Lie algebra b . We first observe: Lemma . Let B = B ′ ⋉ S be a Pyatetskii-Shapiro decomposition with S elementary normal. Then, there exists C > such that for every g = ( a, v, t ) ∈ S , g ′ ∈ B ′ , we have: d B ( g g ′ ) ≥ C (cid:0) d B ′ ( g ′ ) + cosh 2 a + | v | + | t | (cid:1) . Proof.
By Lemma 1.5, we know that there exists C > g ∈ S and g ′ ∈ B ′ , we have, d B ( g g ′ ) ≥ C (cid:0) d B ′ ( g ′ ) + (cid:0) | Ad g R g ′ | + | R g ′− Ad g − | (cid:1) / (cid:1) , In the decomposition R H ⊕ V ⊕ R E of s , the B ′ -action may be expressed under thefollowing matrix form: R g ′ = id A ( g ′ ) 00 0 id , where A ( g ′ ) is the matrix in the linear symplectic group Sp( V , ω ) as defined inProposition 2.9 (iv). From (2.15) and setting as usual g = ( a, v, t ), we thereforeobtain: Ad g R g ′ = id − e a v e a A ( g ′ ) 02 e a t e a ω ( v, A ( g ′ ) . ) e a , R g ′− Ad g − = id A ( g ′ ) − v e − a A ( g ′ ) − − t − e − a ω ( v, . ) e − a . Using once again the equivalence between operator and Hilbert-Schmidt norms on
End ( s ), we deduce for some C > (cid:0) | Ad g R g ′ | + | R g ′− Ad g − | (cid:1) / ≥ C (cid:0) cosh 2 a + | v | cosh a + | t | (cid:1) ≥ C (cid:0) cosh 2 a + | v | + | t | (cid:1) , and the proof follows. (cid:3) Now, we let also b = a ⊕ n be a decomposition of the Lie algebra of a normal j -group B , with n the nilradical of b and a its orthogonal complement. It followsthen that a is an abelian subalgebra, so that b = a ⋉ n and the group B may beidentified to its Lie algebra b with product( a, n ) · ( a ′ , n ′ ) = (cid:0) a + a ′ , ( e − ad a ′ n ) · CBH n ′ (cid:1) , where n · CBH n ′ denotes the Baker-Campbell-Hausdorff series in the Lie algebra n ,which is finite since n is nilpotent. Definition . Let { H j } nj =1 and { N j } mj =1 be bases of a and n respectively.The coordinates system R n + m → a ⊕ n , ( a , . . . , a n , n , . . . , n m ) (cid:0) arcsinh( a ) H + · · · + arcsinh( a n ) H n , n N + · · · + n m N m (cid:1) , are said to be adapted tempered coordinates for B . Lemma . In any adapted tempered coordinates on B , the multiplication andinverse operations are tempered maps R n + m × R n + m → R n + m and R n + m → R n + m respectively. Proof.
Let a , . . . , a n , n , . . . , n m be adapted tempered coordinates on B asin the above definition. Then, sincesinh( a + a ′ ) = sinh a cosh a ′ + cosh a sinh a ′ , the { a i } -coordinates of the multiplication of x, x ′ ∈ R n + m readsinh (cid:0) arcsinh a i + arcsinh a ′ i (cid:1) = a i q a ′ i + a ′ i q a i , so that they clearly are tempered functions in the a i , a ′ i variables. For the n part,recall that there is a decomposition in real root spaces n = L α n α for the adjointaction of a . Now if n ′ ∈ n α , we have e ad (arcsinh( a ) H + ··· +arcsinh( a n ) H n ) n ′ = e α ( H ) arcsinh( a )+ ··· + α ( H ) arcsinh( a n ) n ′ = (cid:18) a + q a (cid:19) α ( H ) . . . (cid:16) a n + p a n (cid:17) α ( H n ) n ′ , which is a tempered function in a , . . . , a n . As the CBH product in a nilpotentgroup is polynomial, linearly decomposing n ′ N + . . . n ′ m N m along the root spacedecomposition and using the above computation, we get that the n i coordinatesof the product of x and x ′ are tempered in all variables. For the inverse map, as( a, n ) − = ( − a, − e − ad a n ), the above computation also shows the result. (cid:3) .4. TEMPERED PAIRS FOR NORMAL j -GROUPS 53 Lemma . Let b = b ′ ⋉ s be a Pyatetskii-Shapiro decomposition of a normal j -algebra b , with s an elementary normal j -algebra and with corresponding Lie groupdecomposition B = B ′ ⋉ S . Denote R : B ′ → Aut ( S ) the associated extensionhomomorphism. Then in any adapted tempered coordinates for B ′ = a ′ ⊕ n ′ , with dim( a ′ ) = n ′ , dim( n ′ ) = m ′ and S = a ⊕ n (recall that by Proposition 2.5, n isan Heisenberg Lie algebra, thus nilpotent), with dim( a ) = 1 , dim( n ) = m , R is atempered map R n ′ + m ′ × R m → R m . Proof.
Let a , . . . , a n ′ , n , . . . , n m ′ and a n ′ +1 , n m ′ +1 , . . . , n m ′ + m be adaptedtempered coordinates for B ′ and S respectively. The group B ′ acts trivially on H n ′ +1 , the generator of a . Moreover, the coordinates a , . . . , a n ′ +1 , n , . . . , n m ′ + m are adapted tempered coordinates for B . Indeed, one knows [ , pages 56-57] thatthe infinitesimal action of H , . . . , H n ′ is real semi-simple with spectrum containedin {− , , } . Denote i ′ : B ′ → B and i : S → B the inclusions seen through thecoordinates. Now by Lemma 2.33, the map( x ′ , x ) ∈ B ′ × S i ′ ( x ′ ) · i ( x ) ∈ B , is tempered. But the n part of that product is exactly R x ′ ( x ) and so, this concludesthe proof. (cid:3) We are now prepared to state and prove the main result of this chapter.
Theorem . Let B be a normal j -group with Pyatetskii-Shapiro decomposi-tion B = (cid:0) S N ⋉ . . . (cid:1) ⋉ S . Parametrizing the elements g, g ′ ∈ B as g = g g . . . g N and g ′ = g ′ g ′ . . . g ′ N with g i , g ′ i ∈ S i , we define S B can : B × B → R , ( g, g ′ ) N X i =1 S S i can ( g i , g ′ i ) , where S S i can is the canonical phase of S i given in Definition 2.15. Then the pair ( B × B , S B can ) is tempered admissible and tame. Proof.
We will use an induction over N , the number of elementary factors in B . Accordingly, we set B = B ′ ⋉ R S , with B ′ := (cid:0) S N ⋉ . . . (cid:1) ⋉ S and S := S . Wethen observe that B × B = ( B ′ × B ′ ) ⋉ R × R ( S × S ) and from Lemma 1.5 and Lemma2.34, that the extension homomorphism R × R =: R is tempered within adaptedcoordinates. By Theorem 2.29, the pair ( S × S , S S can ) is tempered and admissible. Byinduction hypothesis, the latter also holds for ( B ′ × B ′ , S B ′ can ). Moreover, Equations(2.14) tell that, in the “elementary” case of S × S , the adapted tempered coordinatesand the coordinates associated to the phase function are related to one anotherthrough a tempered diffeomorphism. By induction hypothesis, the latter also holdsfor B ′ × B ′ . Obviously, the extension homomorphism R × R is then temperedwithin the coordinates associated to the phase functions as well. Note that underthe parametrization g = g g ′ , h = h h ′ ∈ B ′ , g , h ∈ S , g ′ , h ′ ∈ B ′ ∈ B ′ , themultiplication and inverse maps of B become: gh = g R g ′ ( h ) g ′ h ′ , g − = R g ′− ( g − ) g ′− , and similarly for B × B . From this and the temperedness of the extension homo-morphism R × R , we see that temperedness of the multiplication and inversion lawsin B × B will immediately follow once we will have shown that the map (1.12) is aglobal diffeomorphism from B × B to ( b ⊕ b ) ⋆ . We will return to this question while examining admissibility. To this aim, let us set G := B ′ × B ′ , G := S × S and denoterespectively by g and g their Lie algebras. Let us also set S := S B ′ can , S := S S can ,and let us assume, by induction hypothesis, that the pair ( G , S ) is admissible,with associated decomposition g = ⊕ N k =0 W k . Let us consider an adapted basisof g , { w k } , k = 1 , . . . , dim( g ) with associated coordinates ( b ) k := g w k .S ( b )on G . Similarly, let us consider the basis { w rj | r = 1 , j = 0 , , , } of g adapted to the decomposition (2.11), where, for the values 1 and 2, j consists of amulti-index and where r labels the copies of S in S × S . Accordingly, we have theassociated coordinate system (2.14) on G that now reads ( x ) rj := g w rj .S ( x ).On G := G ⋉ R G , with S ( xb ) := S ( b ) + S ( x ), x ∈ G , b ∈ G , we thencompute that: ( xb ) k := g w k .S ( xb ) = ( b ) k and ( xb ) rj := g w rj .S ( xb ) = ^ R b ( w rj ) .S ( x ) . Hence it suffices to look at the properties of the multipliers within g . FromPyatetskii-Shapiro’s theory, we know that for the values 0 and 3 of j , the action of G is trivial: R b ( w rj ) = w rj . Hence: ( xb ) rj = ( x ) rj , ∀ j ∈ { , } . Hence it suffices to look at the properties of the multipliers within the subspace V × V = V ⊕ V . For j = 1 ,
2, however, the action is not trivial but stabilizescomponent-wise V × V . Accordingly, we set: R b ( w rj ) =: X p =1 [ R b ] pj w rp , ∀ j ∈ { , } , where, again, p is a multi-index. We therefore have:(2.16) ( xb ) rj = X p =1 [ R b ] pj ( x ) rp , ∀ j ∈ { , } . From Pyatetskii-Shapiro’s theory, we know that the linear operator [ R b ] pj is sym-plectic, thus it has jacobian one. In particular, this implies that the map (1.12) is aglobal diffeomorphism from G to g ⋆ . (Hence, we have completed the proof of tem-peredness of the pair ( B × B , S B can ).) We now consider the ordered decomposition: g = g ⊕ g = (cid:16) ⊕ j =0 V j (cid:17) M (cid:16) ⊕ N k =0 W k (cid:17) , where indices occurring on the left ( g ) are considered as lower than the one on theright ( g ). Within this setting, we compute that for every element X ∈ U ( g ): (2) α X ( xb ) := e − iS ( xb ) (cid:0) e X e iS (cid:1) ( xb ) = α R b ( X ) ( x ) , where α X := e − iS (cid:0) e X.e iS (cid:1) denotes the multiplier on G . Again, for the extremevalues of j , we observe that: (2) α X ( xb ) = α X ( x ) , ∀ X ∈ U ( V ⊕ V ) , so in these cases the properties underlying admissibility are trivially satisfied. For j = 1 ,
2, we have with the notation X j := 1 − P r =1 ( w rj ) of Proposition 2.26: R b ( X j ) = 1 − X r =1 (cid:0) [ R b ] pj w rp (cid:1) , .4. TEMPERED PAIRS FOR NORMAL j -GROUPS 55 which leads to (2) α X j ( xb ) = 1 + X r =1 (cid:0) ( xb ) rj (cid:1) − i X r =1 [ R b r ] p r j [ R b r ] p ′ r j g w rp r . ( x ) rp ′ r , where b = ( b , b ) ∈ B ′ × B ′ = G . This gives the property ( i ) of Definition 1.24.From the expression (2.16) and the structure of the elementary case (Lemma 2.20),we then observe: e A. (2) α X j = 0 , ∀ A ∈ U ( V × V ) , deg( A ) ≥ . (2.17)Also, setting − iβ X j ( xb ) := − i P r =1 [ R b r ] p r j [ R b r ] p ′ r j g w rp r . ( x ) rp ′ r , we deduce fromthe expressions (2.8) and (2.9) that, for every A ∈ V × V : e A.β X j = 0. From theexpression (2.16) and setting ( x ) := ( ( x ) , ( x ) ), we then deduce that for every A ∈ U ( V × V ) of strictly positive degree: (cid:12)(cid:12) e A (cid:0) (2) α X ( xb ) + (2) α X ( xb ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) e A | ( xb ) | (cid:12)(cid:12) = (cid:12)(cid:12) ^ R b ( A ) x | R b ( ( x ) ) | (cid:12)(cid:12) ≤ | R b | deg( A )+2 C A µ ( x ) | ( ( x ) ) | , where the last estimate is obtained from Corollary 2.24. Since | ( ( x ) ) | = | R b − R b ( ( x ) ) | ≤ | R b − | | ( ( xb ) ) | , we then get (cid:12)(cid:12) e A (cid:0) (2) α X ( xb ) + (2) α X ( xb ) (cid:1)(cid:12)(cid:12) ≤ | R b | deg( A )+2 C A µ ( x ) | R b − | | ( ( xb ) ) | ≤ C A | R b | deg( A )+2 | R b − | µ ( x ) (cid:12)(cid:12) (2) α X ( xb ) + (2) α X ( xb ) (cid:12)(cid:12) . But by (2.17), we know that we may assume deg( A ) ≤
2, hence (cid:12)(cid:12) e A (cid:0) (2) α X ( xb ) + (2) α X ( xb ) (cid:1)(cid:12)(cid:12) ≤ C A d G ( b ) µ ( x ) (cid:12)(cid:12) (2) α X ( xb ) + (2) α X ( xb ) (cid:12)(cid:12) . Defining the element µ ′ ( xb ) := d G ( b ) µ ( x ) yields admissibility at the level of V × V .Last, tameness also follows by an induction argument, using Lemma 1.5 andLemma 2.31. (cid:3) Remark . Observe that Remarks 1.20, Lemma 1.21 and Lemma 2.31 showthat on the one-variable Schwartz space S S B can ( B , E ) associated with the two-variabletempered, admissible and tame pair ( B × B , S B can ) (see Definition 1.48) one has thesame topology associated with the equivalent semi-norms f ∈ S S ( B , E ) sup X ∈ U k ( b ) sup x ∈ B n d B ( x ) n (cid:13)(cid:13) e X f ( x ) (cid:13)(cid:13) j | X | k o , j, k, n ∈ N , or even with f ∈ S S ( B , E ) sup X ∈ U k ( b ) sup x ∈ B n d B ( x ) n (cid:13)(cid:13) X f ( x ) (cid:13)(cid:13) j | X | k o , j, k, n ∈ N . HAPTER 3
Non-formal star-products
This chapter is devoted to the construction of an infinite dimensional parameterfamily of non-formal star-products on every negatively curved K¨ahlerian Lie group.In section 3.1, we first review the construction of one of us of such non-formalstar-products, living on a space of distributions. Their covariance properties andtheir relations with the Moyal product are given there. In section 3.2, we applythe results of chapters 1 and 2 to give a proper interpretation of the star-productformula as an oscillatory integral and eventually prove that the Fr´echet space B ( B )becomes a Fr´echet algebra for all these star-products. This is proved in Theorem3.9, the main result of this chapter and, we believe, and important result in itself. We consider an elementary normal j -group S viewed as a symplectic symmetricspaces as in section 2.2. We start by recalling the results obtained in [ , ]. Definition . Set ˜ S := { ( a, v, ξ ) } = R × R d × R . The twisting map is thesmooth one-parameter family of diffeomorphisms defined as φ θ : ˜ S → ˜ S : ( a, v, ξ ) (cid:16) a, sech (cid:0) θ ξ (cid:1) v, θ sinh (cid:0) θ ξ (cid:1) (cid:17) , θ ∈ R ∗ . Let S ( S ) be the Euclidean Schwartz space of S , i.e . the ordinary Schwartz spacein the coordinates (2.4). Accordingly, let S ( S ) ′ be the dual space of tempereddistributions. Let us also denote by (cid:0) F u (cid:1) ( a, v, ξ ) := Z ∞−∞ e − iξt u ( a, v, t ) d t , the partial Fourier transform in the t -variable. For γ >
0, we let O C,γ ( R m ) be thesubset of smooth functions, the derivatives of which are uniformly polynomiallybounded: O C,γ ( R m ) := (cid:8) f ∈ C ∞ ( R m ) : ∃ r > ∀ α ∈ N m : ∃ C α > , | ∂ α f ( x ) | ≤ C α (1 + | x | ) r − γ | α | (cid:9) . Note that O C, ( R m ) is the space of Grossmann-Loupias-Stein symbols, traditionallywritten K ( R m ). Definition . We denote by Θ , the subspace of C ∞ ( R , C ) constituted bythe elements τ such that exp ◦ ± τ belong to the space O C, ( R , C ), and normalizedsuch that τ (0) = 0.Let τ be the element of C ∞ (˜ S ), given by: τ := log ◦ Jac φ − θ .
578 3. NON-FORMAL STAR-PRODUCTS
Viewed as a function of its last variable only, τ belongs to Θ . Indeed, we have:Jac φ − θ ( a, v, ξ ) = 2 − d (cid:16) q θ ξ (cid:17) d q θ ξ . To an element τ ∈ Θ , one associates a function on ˜ S by [( a, v, ξ ) τ ( ξ )] and,to simplify the notations, we still denote this function by τ . One then defines alinear injection:(3.1) T θ,τ := F − ◦ exp( τ − τ ) ◦ ( φ − θ ) ⋆ ◦ F : S ( S ) → S ( S ) ′ , where, by a slight abuse of notation, we identify a function with the linear operatorof point-wise multiplication by this function. We make the following obvious butimportant observation: Lemma . Let τ ∈ Θ . Then the inverse of the map T θ,τ , given by T − θ,τ = F − ◦ ( φ θ ) ⋆ ◦ exp( − τ + τ ) ◦ F , defines a continuous linear injection from S ( S ) to itself. Moreover, T − θ,τ extends toa unitary operator on L ( S ) if and only if τ is purely imaginary. Proof.
As the partial Fourier transform F is an homeomorphism from S ( S )to S (˜ S ), and as exp( − τ + τ ) belongs to O C, ( R , C ), a subspace of the Schwartzmultipliers O M ( R , C ), it suffices to show that the map f f ◦ φ θ , is continuouson S (˜ S ). So, let f ∈ S (˜ S ). Then, as f ◦ φ θ ( a, v, ξ ) = f (cid:0) a, sech (cid:0) θ ξ (cid:1) v, θ sinh (cid:0) θ ξ (cid:1) (cid:1) , we deduce from ∂ ka ( f ◦ φ θ ) = ( ∂ ka f ) ◦ φ θ , ∂ mv ( f ◦ φ θ ) = sech (cid:0) θ ξ (cid:1) m ( ∂ mv f ) ◦ φ θ , and from an iterative use of the relation ∂ ξ ( f ◦ φ θ )( a, v, ξ ) = − θ (cid:0) θ ξ (cid:1) cosh (cid:0) θ ξ (cid:1) v ( ∂ v f ) ◦ φ θ ( a, v, ξ )+cosh (cid:0) θ ξ (cid:1) ( ∂ ξ f ) ◦ φ θ ( a, v, ξ ) , that any (ordinary) derivatives of f ◦ φ θ is a finite linear combination of derivativesof f composed with φ θ and with coefficient in the polynomial ring R [ v, e θξ , e − θξ ].To conclude the first claim, we then observe that as a derivative of a Schwartzfunction is a Schwartz function, any derivatives of f composed with φ θ is bounded(in absolute value) by ( a + | v | + cosh( θξ )) − n , with n arbitrary. For the secondclaim, observe that T − θ,τ = F − ◦ ( φ θ ) ⋆ ◦ Jac − / φ − θ ◦ exp( τ ) ◦ F = F − ◦ Jac / φ θ ◦ ( φ θ ) ⋆ ◦ exp( τ ) ◦ F , which entails since F and Jac / φ θ ◦ ( φ θ ) ⋆ are unitary, that T − θ,τ is unitary if andonly if the operator of multiplication by exp( τ ) is unitary, which is equivalent to ℜ ( τ ) = 0. (cid:3) Remark . Denoting by T ∗ θ,τ the formal adjoint of T θ,τ with respect to theinner product of L ( S ), then we have T ∗ θ,τ = T − θ, − τ and thus, Lemma (3.3) impliesthat T ∗ θ,τ is continuous on S ( S ) too. .1. STAR-PRODUCTS ON NORMAL j -GROUPS 59 Let ω be the standard symplectic structure of R d +2 ≃ T ∗ R d +1 and let ⋆ θ bethe Moyal product in its integral form on S ( R d +2 ). Recall that the latter productis by definition the composition law of symbols in the Weyl pseudo-differentialcalculus and that it is given by f ⋆ θ f ( x ) = 1( πθ ) d +1) Z R d +2 × R d +2 e iθ S ( x,y,z ) f ( y ) f ( z ) d y d z , where S ( x, y, z ) := ω ( x, y ) + ω ( y, z ) + ω ( z, x ). For τ ∈ Θ , denoting by E θ,τ ( S ) := T θ,τ (cid:0) S ( S ) (cid:1) , the range subspace of T θ,τ in the tempered distribution space S ( S ) ′ , one has theinclusions S ( S ) ⊂ E θ,τ ( S ) ⊂ C ∞ ( S ) . We consider the linear isomorphism: T − θ,τ : E θ,τ ( S ) → S ( S ) . Identifying S ≃ R d +2 by means of the global coordinate system (2.4), we transportunder T θ,τ the Moyal product on S ( R d +2 ) ≃ S ( S ). This yields an associativeproduct: ⋆ θ,τ : E θ,τ ( S ) × E θ,τ ( S ) → E θ,τ ( S ) , given by(3.2) f ⋆ θ,τ f := T θ,τ (cid:0) T − θ,τ ( f ) ⋆ θ T − θ,τ ( f ) (cid:1) , f , f ∈ E θ,τ ( S ) . The associative algebra (cid:0) E θ,τ ( S ) , ⋆ θ,τ (cid:1) , endowed with the Fr´echet algebra structuretransported under T θ,τ from S ( R d +2 ), satisfies the following properties [ , ]: Theorem . Let τ ∈ Θ and θ = 0 . Then, (i) For all compactly supported u, v ∈ E θ,τ ( S ) , one has the integral represen-tation: (3.3) u ⋆ θ,τ v = Z S × S K θ,τ ( x , x ) R ⋆x ( u ) R ⋆x ( v ) d S ( x ) d S ( x ) , where the two-point kernel is given by K θ,τ ( x , x ) := ( πθ ) − d +1) A θ,τ ( x , x ) exp (cid:8) iθ S S can ( x , x ) (cid:9) , (3.4) with, in the coordinates (2.4) : A θ,τ ( x , x ) := A S can ( x , x ) exp (cid:8) τ (cid:0) θ sinh 2 a (cid:1) + τ (cid:0) − θ sinh 2 a (cid:1) − τ (cid:0) θ sinh(2 a − a ) (cid:1)(cid:9) , and with S S can and A S can defined in (2.6) and (2.7) . (ii) The product ⋆ θ,τ is equivariant under the automorphism group of the sym-plectic symmetric space ( S , s, ω S ) : for all elements g of Aut ( S , s, ω S ) and u, v ∈ D ( S ) , one has g ⋆ ( u ) ⋆ θ,τ g ⋆ ( v ) = g ⋆ ( u ⋆ θ,τ v ) . As usual, we set x j = ( a j , v j , t j ) ∈ R d +2 Remark . Observe that when ℑ ( τ ) = 0, then the amplitude A θ,τ alsocontribute to the phase of K θ,τ . However, as the amplitude is a function of thevariables in a × a := R H × R H only, in the two dimensional case (i.e . V = { } ),there is no τ ∈ Θ such that the combined phase coincides with those found in [ ].In particular, this means that the upper half-plane is a symplectic manifold thatsupports two non-isomorphic Moyal quantizations.Consider a normal j -group decomposed, following Proposition 2.5, into a semi-direct product B = B ′ ⋉ S where S is elementary. One knows from Proposition 2.5and [ ] that the extension homomorphism R : B ′ → Aut ( S ) underlies a homomor-phism from B ′ into the isotropy subgroup Aut ( S , s, ω S ) e at the unit element e of S viewed as a symmetric space: R : B ′ → Sp(
V, ω ) ⊂ Aut ( S , s, ω S ) e , where ( V, ω ) is the symplectic vector space attached to S . In particular, the actionof B ′ leaves invariant the two-point kernel K θ,τ on S × S . Iterating the aboveobservation at the level of B ′ and translating the “extension Lemma” in [ ] withinthe present framework, we obtain: Proposition . Let B be a normal j -group with Pyatetskii-Shapiro decompo-sition B = ( S N ⋉ . . . ) ⋉ S and fix ~τ := ( τ , . . . , τ N ) ∈ Θ N . Parametrizing a groupelement g ∈ B as g = g . . . g N , with g i ∈ S i , we consider the two-point kernel on B given by (3.5) K θ,~τ ( g, g ′ ) := K θ,τ ( g , g ′ ) . . . K θ,τ N ( g N , g ′ N ) , where K θ,τ i is the two-points kernel on S i × S i , defined in (3.4) . Then, the bilinearmapping ⋆ θ,~τ := h ( u, v ) Z B × B K θ,~τ ( g, g ′ ) R ⋆g ( u ) R ⋆g ′ ( v ) d B ( g ) d B ( g ′ ) i , is associative on E θ,~τ ( B ) := E θ,τ N ( S N ) ⊗ · · · ⊗ E θ,τ ( S ) , (recall that E θ,τ j ( S j ) is nuclear). Moreover, at the level of compactly supportedfunctions, the product ⋆ θ,~τ is equivariant under the left-translations in B . In this section, we fix B a normal j -group, with Lie algebra b . We also let ~τ ∈ Θ N be as above ( N is the number of elementary components in B ) and formthe two-point kernel K θ,~τ on B × B , defined in (3.5). Proposition 3.7 implies thatthe deformed product(3.6) u ⋆ θ,~τ v = Z B × B K θ,~τ ( g, g ′ ) R ⋆g ( u ) R ⋆g ′ ( v ) d B ( g ) d B ( g ′ ) , is weakly associative (in the sense of Definition 1.46) and left B -equivariant. Theresults of chapter 1 will allow to properly understand the integral in (3.6) as anoscillatory one. As a consequence, we will see that the deformed product extendsas a continuous bilinear and associative map on the function space B ( B , A ), for A a Fr´echet algebra. We start with a simple fact: .2. AN OSCILLATORY INTEGRAL FORMULA FOR THE STAR-PRODUCT 61 Lemma . Let B be an elementary normal j -group and ~τ ∈ Θ N . Then theamplitude A θ,~τ , as given in Proposition 3.7, consists of an element of B µ τ ( B × B ) for a tempered weight µ τ . Proof.
Consider first the case where B = S is elementary. Within the nota-tions of section 2.3, we have | ~x | = | ( x , , x , ) | = (cid:12)(cid:12)(cid:0) − sinh 2 a , sinh 2 a (cid:1)(cid:12)(cid:12) = (cid:0) sinh a + sinh a (cid:1) / , so that the function µ can ( x , x ) := cosh a cosh a , is a tempered weight. As the left invariant vector field e H on S restricted to func-tions of depending on the variable a only, coincides with the partial differentiationoperator ∂ a , we get from the explicit expression A S can ( x , x ) = (cosh a cosh a cosh( a − a )) d p cosh 2 a cosh 2 a cosh 2( a − a ) , that there exists ρ > X ∈ U ( s ⊕ s ), there exists a constant C X > (cid:12)(cid:12) e X A S can (cid:12)(cid:12) ≤ C X µ ρ can . Hence A S can ∈ B µ ρ can ( S × S ). Next, since τ ∈ Θ , we have exp ◦ ± τ ∈ O C, ( R ).Thus, there exists r > n -th derivative of exp ◦ ± τ ( x ) is boundedby (1 + | x | ) r − n . Let us denote by deg( τ ) such positive number r . Since exp ◦ ± τ depends on the variable a only, among all elements of U ( s ⊕ s ), only the powers of e H i , i = 1 ,
2, give non zero contributions. Therefore, an easy computation showsthat for any X ∈ U ( s ⊕ s ), there exists a constant C X > (cid:12)(cid:12) e X exp {± τ (cid:0) θ sinh 2 a (cid:1) } (cid:12)(cid:12) ≤ C X (1 + | ~x | ) τ ) . Hence A θ,τ belongs to B µ τ ( S × S ) for µ τ = µ ρ +3 deg( τ )can .The general case B = B ′ ⋉ S follows easily by Pyatetskii-Shapiro theory, sinceonly the variables in V ⊂ S are affected by the action of B ′ and that A θ,~τ isindependent of these variables. (cid:3) We now consider a Fr´echet algebra A , with topology underlying a countablefamily of sub-multiplicative semi-norms {k . k j } j ∈ N . Combining Lemma 3.8 withTheorem 2.35 leads us to prove that the integral in the expression of the deformedproduct (3.3) can be properly understood as an oscillatory one in the sense ofchapter 1. In particular, this allows to define the product ⋆ θ,~τ on B ( B , A ). This isthe main result of this chapter.Before going further, a clarification regarding the notion of temperedness shouldbe made. Recall that in the framework of a tempered pair ( G, S ), the notion oftemperedness comes from the global coordinate system associated to the phasefunction S (see Definition 1.17 and Definition 1.22). Thus, in the case of a normal j -group B , this notion is a priori only defined on the product group B × B . However, wehave seen in the proof of Theorem 2.35 that the coordinate system (1.12) associatedto the phase function S B can is related to the adapted tempered coordinates on B × B (see Definition 2.32) by a tempered diffeomorphism. Hence, it seems natural todefine directly the notion of temperedness on B by mean of the adapted temperedcoordinates. Having saying that, the important observation is that two functions ϕ , ϕ are tempered on B , if and only if ϕ ⊗ ϕ is tempered on B × B . It impliesa great simplification of the assumptions in Theorem 1.43, Proposition 1.47 and Proposition 1.50, namely the temperedness at the level of tensor product of weightscan be reduced to temperedness at the level of each factor.
Theorem . Let B be a normal j -group. Fix ~τ ∈ Θ N and let µ , µ , µ bethree families of tempered (in the sense of adapted tempered coordinates) weightson B of sub-multiplicativity degree ( L , R ) , ( L , R ) , ( L , R ) . Considering K θ,~τ the two-point kernel on B defined in (3.5) , the correspondence ⋆ θ,~τ : B µ ( B , A ) × B µ ( B , A ) → B ν ( B , A ) with ν j = µ L ,j ,j µ L ,j ,j , ( F , F ) ^ Z B × B K θ,~τ h ( x , x ) R ⋆x ( F ) R ⋆x ( F ) i , is a continuous bilinear map and is equivariant under the left translations in B inthe sense that for all g ∈ B , we have L ⋆g ( F ⋆ θ,~τ F ) = ( L ⋆g F ) ⋆ θ,~τ ( L ⋆g F ) in B λ ( B , A ) for λ = { µ L ,j R ,j ,j µ L ,j R ,j ,j } . Moreover, the map ⋆ θ,~τ is associative in the sense that then for every elements F j in B µ j ( B , A ) , j = 1 , , , we have the equality (cid:0) F ⋆ θ,~τ F (cid:1) ⋆ θ,~τ F = F ⋆ θ,~τ (cid:0) F ⋆ θ,~τ F (cid:1) in B ρ ( B , A ) , with ρ = { µ L ,j ,j µ ,jL ,j µ ,jL ,j } j ∈ N . In particular, (cid:0) B ( B , A ) , ⋆ θ,~τ (cid:1) is a Fr´echet algebra. Proof.
That the bilinear map ⋆ θ,~τ (with the domain and image as indicated)is well defined and continuous, follows from Theorem 1.43 (cf . the above discus-sion for the condition of temperedness of the weights involved), Theorem 2.35 andLemma 3.8. Associativity follows from associativity in E θ,~τ ( B ), which implies weakassociativity in the sense of Definition 1.46 and Proposition 1.47. So, it remains toprove left B -equivariance. We first note that by Lemma 1.12 (ii), the group B actson the left continuously from B µ ( B , A ) to B γ ( B , A ), with γ = { µ R j j } (for any familyof weights µ of sub-multiplicative degree ( L, R )). Also, we have by Lemma 1.44that F ⋆ θ,~τ F = lim n ,n F ,n ⋆ θ,~τ F ,n in B ν ( B , A ), with ν = { µ L ,j ,j µ L ,j ,j } , forany pair of sequences { F ,n } and { F ,n } of smooth compactly supported A -valuedfunctions on B , which converge to F and F , in the topology of B ˆ µ ( B , A ) and B ˆ µ ( B , A ) for any sequence of weights ˆ µ and ˆ µ dominating µ and µ . Fromcontinuity of the left regular action (see Lemma 1.12 ii) and left B -equivariance atthe level of D ( B , A ), we thus have L ⋆g (cid:0) F ⋆ θ,~τ F ′ (cid:1) = lim n,n ′ →∞ L ⋆g (cid:0) F n ⋆ θ,~τ F ′ n ′ (cid:1) = lim n,n ′ →∞ ( L ⋆g F n ) ⋆ θ,~τ ( L ⋆g F ′ n ′ ) , where the limits are in B λ ( B , A ), for λ = { µ L ,j R ,j ,j µ L ,j R ,j ,j } . It remains to find spe-cific approximation sequences { F ,n } and { F ,n } , such that { L ⋆g F ,n } and { L ⋆g F ,n } converge to L ⋆g F and L ⋆g F , in the topology of B ˆ γ ( B , A ) and B ˆ γ ( B , A ) withˆ γ = { ˆ µ R ,j ,j } and ˆ γ = { ˆ µ R ,j ,j } . For this, we observe that the same construc-tion as in the proof of Lemma 1.8 (viii), does the job. Indeed, recall that there, wehave constructed the approximation sequence { F n } , by setting for F ∈ B ( B , A ): F n := e n F ∈ D ( B , A ) where e n := Z B ψ ( g ) R ⋆g ( χ C n ) d B ( g ) ∈ D ( B ) , .2. AN OSCILLATORY INTEGRAL FORMULA FOR THE STAR-PRODUCT 63 and 0 ≤ ψ ∈ D ( B ), R B ψ ( x ) d B ( x ) = 1, { C n } is an increasing sequence of relativelycompact open subsets of B converging to B and χ C n is the characteristic function of C n . Fixing g ∈ B and setting C gn := g.C n , the sequence { C gn } is still an increasingsequence of relatively compact open subsets on B converging to B . Also, as e gn := L ⋆g ( e n ) = Z B ψ ( g ′ ) R ⋆g ′ ( χ C gn ) d B ( g ′ ) ∈ D ( B ) , we deduce that for all j, k ∈ N : k L ∗ g ( F n ) − L ∗ g ( F ) k j,k,γ = k (1 − e gn ) L ∗ g ( F ) k j,k,γ with ˆ γ = { ˆ µ R j j } j ∈ N , which, by Lemma 1.12 (vi), converges to zero as L ∗ g ( F ) ∈ B γ ( B , A ) with γ = { µ R j j } and γ ≺ ˆ γ . (cid:3) Let S S B can ( B , A ) be the one-variable Schwartz space associated to the admissibleand tame tempered pair ( B × B , S B can ), constructed in Definition 1.48. The nextresult follows immediately from Proposition 1.50. Proposition . Let B be a normal j -group and ~τ ∈ Θ N . Then, endowedwith the multiplication ⋆ θ,~τ , the space S S B can ( B , A ) becomes a Fr´echet algebra which,for µ an arbitrary family of tempered weights, acts continuously on B µ ( B , A ) , via L ⋆ θ,~τ ( F ) : ϕ F ⋆ θ,~τ ϕ , F ∈ B µ ( B , A ) , ϕ ∈ S S B can ( B , A ) . In particular, (cid:0) S S B can ( B , A ) , ⋆ θ,~τ (cid:1) is an ideal of (cid:0) B ( B , A ) , ⋆ θ,~τ (cid:1) . We now see that, as expected, the constant function is an identity for thedeformed product.
Proposition . Let B be a normal j -group. Fix ~τ ∈ Θ N , µ a family oftempered weights of sub-multiplicative degree ( L , R ) and F ∈ B µ ( B , A ) . Identifyingan element a ∈ A with the function [ g a ] in B ( B , A ) , we have a ⋆ θ,~τ F = aF , F ⋆ θ,~τ a = F a , in B µ ( B , A ) . In particular, if A is unital, the element [ g A ] ∈ B ( B , A ) is theunit of (cid:0) B ( B , A ) , ⋆ θ,~τ (cid:1) . Proof.
Since the constant unit function is a fixed point of the map T − θ,~τ , forevery ϕ ∈ S S B can ( B , A ), we have: ϕ ⋆ θ,~τ a = T θ,~τ (cid:0) T − θ,~τ ( ϕ ) ⋆ θ a (cid:1) , in S S B can ( B , A ). By Remark 2.36, we see that the transported Schwartz space S S B can ( B , A ) is a (dense) subset of the ordinary Schwartz space S ( b , A ), underthe usual identification B ≃ b . Since T − θ,~τ preserves the latter space, we seethat T − θ,~τ ( ϕ ) ∈ S ( b , A ). It is well known (see [ ] for the Fr´echet algebra val-ued case) that the Moyal product admits the constant function as unit element.Thus ϕ ⋆ θ,~τ a = ϕa and a ⋆ θ,~τ ϕ = aϕ for all ϕ ∈ S S B can ( B , A ) and a ∈ A . Now,consider the injective homomorphism L ⋆ θ,~τ from (cid:0) B µ ( B , A ) , ⋆ θ,~τ (cid:1) to the algebra ofcontinuous operators acting on S S B can ( B , A ), defined in Proposition 3.10. From theprevious considerations, the associativity of the deformed product and the fact that S S B can ( B , A ) is an ideal of B µ ( B , A ), we get L ⋆ θ,~τ ( F ⋆ θ,~τ a ) = L ⋆ θ,~τ ( F a ) , ∀ F ∈ B µ ( B , A ) , which entails by injectivity that F ⋆ θ,~τ a = F a in B ν ( B , A ), with ν = { µ L j j } . As F a ∈ B µ ( B , A ), we deduce that the equality F ⋆ θ,~τ a = F a holds in fact in B µ ( B , A ).The case of a ⋆ θ,~τ F is entirely similar. (cid:3) HAPTER 4
Deformation of Fr´echet algebras
In this chapter, we consider a normal j -group B and a pair ( A , α ), consisting ofa Fr´echet algebra A , together with a strongly continuous (not necessarily isometric)action α of B by automorphisms. For a ∈ A , we let α ( a ) be the A -valued functionon B , defined by(4.1) α ( a ) := [ g ∈ B α g ( a ) ∈ A ] . Our main goal, achieved in section 4.1, is to show that the formula a ⋆ αθ,~τ b := (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) ( e ) , equips A ∞ with a new noncommutative and associative Fr´echet algebra structure.This is proved in Theorem 4.8 as a direct consequence of Theorem 3.9 and of Lemma4.5 which shows that the map (4.1) is continuous from A ∞ to B µ ( B , A ∞ ), for µ anontrivial family of tempered weights on the group B . We do not yet need to work with isometric actions. We start by general con-siderations regarding tempered actions:
Definition . A tempered action of a tempered Lie group G on a Fr´echetalgebra A , is given by the data ( α, µ α ) where α is an action of G on A and µ α is afamily of tempered weights on G such that for all j ∈ N , all a ∈ A and all g ∈ G ,we have k α g ( a ) k j ≤ µ αj ( g ) k a k j . Remark . Note that for a tempered action and for g ∈ G fixed, α g actscontinuously on A .We denote by A ∞ the set of smooth vectors for the action α of B on A : A ∞ := (cid:8) a ∈ A : α ( a ) ∈ C ∞ ( B , A ) (cid:9) . When the action is strongly continuous, A ∞ is a dense subspace of A . On thissubset, we consider the infinitesimal form of the action, given for X ∈ b by: X α ( a ) := ddt (cid:12)(cid:12)(cid:12) t =0 α e tX ( a ) , a ∈ A ∞ , and extended to the whole universal enveloping algebra U ( b ), by declaring that themap U ( b ) → End ( A ∞ ), X X α is an algebra homomorphism. The subspace A ∞ carries a finer topology associated with the following set of semi-norms: k a k j,X := k X α ( a ) k j , a ∈ A ∞ , X ∈ U ( b ) , j ∈ N . Considering the PBW basis of U ( b ) associated to an ordered basis of b as in (0.4),one can use only countably many semi-norms to define the topology of A ∞ . The
656 4. DEFORMATION OF FR´ECHET ALGEBRAS latter are indexed by ( j, k ) ∈ N , where j refers to the labeling of the initial familyof semi-norms {k . k j } j ∈ N of A and k refers to the labeling of the filtration U ( b ) = ∪ k ∈ N U k ( b ) associated to the chosen PBW basis, as defined in (0.5). In turn, A ∞ becomes a Fr´echet space, for the topology associated with the semi-norms(4.2) k . k j,k : A ∞ → [0 , ∞ ) , a sup X ∈ U k ( g ) k a k j,X | X | k = sup X ∈ U k ( g ) k X α ( a ) k j | X | k , with j, k ∈ N and where | . | k is the ℓ -norm of U k ( b ) defined in (0.7). As in (1.4),we have k a k j,k ≤ max | β |≤ k k a k j,X β , with { X β , | β | ≤ k } the basis (0.4) of U k ( b ). Hence the semi-norms (4.2) are welldefined on A ∞ .In the context of a tempered action on a Fr´echet algebra A , we observe thatthe restriction of the action to A ∞ is also tempered, but never isometric, even ifthe action is isometric on A unless the group is Abelian. This explains why in ourcontext it is natural to work with tempered actions, rather than with isometricones. Lemma . Let ( A , α, µ α ) be a Fr´echet algebra endowed with a tempered actionof a tempered Lie group G . Then, the restriction of α on A ∞ is tempered too, with: k α g ( a ) k j,k ≤ C ( k ) d G ( g ) k µ αj ( g ) k a k j,k , j, k ∈ N , g ∈ G, a ∈ A ∞ . Proof.
First remark k α g ( a ) k j,k = sup X ∈ U k ( g ) k α g (cid:0)(cid:0) Ad g − ( X ) (cid:1) α ( a ) (cid:1) k j | X | k ≤ µ αj ( g ) sup X ∈ U k ( g ) k (cid:0) Ad g − ( X ) (cid:1) α ( a ) k j | X | k . As for X ∈ U k ( g ) and a ∈ A ∞ , we have k X α ( a ) k j ≤ | X | k sup Y ∈ U k ( g ) k Y α ( a ) k j | Y | k = | X | k k a k j,k , we get, with | Ad g | k denoting the operator norm of the adjoint action of G on thenormed space (cid:0) U k ( g ) , | . | k (cid:1) : k α g ( a ) k j,k ≤ µ αj ( g ) sup X ∈ U k ( g ) | Ad g − ( X ) | k | X | k k a k j,k = µ αj ( g ) | Ad g − | k k a k j,k , and one concludes using Lemma 1.14. (cid:3) Example . Applying the former result to α = R ⋆ and A = S S B can ( B ) (whichis its own space of smooth vectors), we see that the right-action of B on S S B can ( B )is tempered.The following statement is the foundation of our construction: Lemma . Let ( α, µ α ) be a tempered and strongly continuous action of aLie group G on a Fr´echet algebra A . Setting then ν := { µ αj d kG } j,k ∈ N , we have anequivariant continuous embedding α : A ∞ → B ν ( G, A ∞ ) , a α ( a ) = [ g ∈ G α g ( a ) ∈ A ∞ ] . .1. THE DEFORMED PRODUCT 67 Proof.
Note first that for a ∈ A and g, g ∈ G , we have α (cid:0) α g ( a ) (cid:1) ( g ) = α g g ( a ) = (cid:0) R ⋆g α ( a ) (cid:1) ( g ) , and thus α : a ∈ A 7→ [ g α g ( a )] ∈ C ( G, A ) intertwines the actions R ⋆ and α .Let now a ∈ A ∞ and X ∈ U ( g ). By equivariance and strong-differentiability of α on A ∞ , we get e Xα ( a ) = α ( X α a ) . Since for all j ∈ N and all a ∈ A , we have k α g ( a ) k j ≤ µ αj ( g ) k a k j , we deduce that k α ( a ) k j,k,µ α = sup X ∈ U k ( g ) sup g ∈ G k e Xα g ( a ) k j µ αj ( g ) | X | k = sup X ∈ U k ( g ) sup g ∈ G k α g ( X α a ) k j µ αj ( g ) | X | k ≤ sup X ∈ U k ( g ) k X α a k j | X | k = k a k j,k . This analysis shows that the map α : A ∞ → B µ α ( G, A ) is continuous. Now wewant to take into account the intrinsic topology of A ∞ in the target space of themap α . Remark that the topology of B ν ( G, A ∞ ) is associated with the countableset of semi-norms k F k ( j,k ) ,k ′ ,ν = sup X ∈ U k ′ ( g ) sup g ∈ G sup Y ∈ U k ( g ) k Y α (cid:0) e XF ( g ) (cid:1) k j ν j,k ( g ) | X | k ′ | Y | k . Since α g − ◦ X α ◦ α g = ( Ad g − X ) α for all X ∈ U ( g ) and g ∈ G , we get for F = α ( a )and ν = { µ αj d kG } j,k ∈ N : k α ( a ) k ( j,k ) ,k ′ ,ν = sup X ∈ U k ′ ( g ) sup g ∈ G sup Y ∈ U k ( g ) k Y α (cid:0) e X g α g ( a ) (cid:1) k j µ αj ( g ) d G ( g ) k | X | k ′ | Y | k = sup X ∈ U k ′ ( g ) sup g ∈ G sup Y ∈ U k ( g ) k Y α (cid:0) α g ( X α a ) (cid:1) k j µ αj ( g ) d G ( g ) k | X | k ′ | Y | k = sup X ∈ U k ′ ( g ) sup g ∈ G sup Y ∈ U k ( g ) k α g (cid:0) ( Ad g − Y ) α X α a (cid:1) k j µ αj ( g ) d G ( g ) k | X | k ′ | Y | k ≤ sup X ∈ U k ′ ( g ) sup g ∈ G sup Y ∈ U k ( g ) k ( Ad g − Y ) α X α a k j d G ( g ) k | X | k ′ | Y | k ≤ (cid:16) sup g ∈ G | Ad g − | k d G ( g ) k (cid:17) sup X ∈ U k ′ ( g ) sup Y ∈ U k ( g ) k Y α X α a k j | X | k ′ | Y | k ≤ (cid:16) sup g ∈ G | Ad g − | k d G ( g ) k (cid:17)(cid:16) sup X ∈ U k ′ ( g ) sup Y ∈ U k ( g ) | Y X | k + k ′ | X | k ′ | Y | k (cid:17) sup Z ∈ U k + k ′ ( g ) k Z α a k j | Z | k + k ′ = (cid:16) sup g ∈ G | Ad g − | k d G ( g ) k (cid:17)(cid:16) sup X ∈ U k ′ ( g ) sup Y ∈ U k ( g ) | Y X | k + k ′ | X | k ′ | Y | k (cid:17) k a k j,k + k ′ , and one concludes using Lemma 1.14. (cid:3) The next result, although rather obvious, will also play a key role.
Lemma . Let A be a Fr´echet algebra and let µ be a family of temperedweights on a tempered Lie group G . Then, the evaluation map at the unit element, B µ ( G, A ) → A , F F ( e ) , is continuous. Proof.
Fix j ∈ N . We have for any F ∈ B µ ( G, A ): k F ( e ) k j ≤ µ j ( e ) sup g ∈ G k F ( g ) k j µ j ( g ) = µ j ( e ) k F k j, ,µ , and the result follows immediately. (cid:3) Last, we need to lift the action α from A ∞ to B ν ( B , A ∞ ), ν = { µ αj d kG } , and toshow that this lift acts by automorphisms of the product ⋆ θ,~τ . Lemma . Let ( α, µ α ) be a strongly continuous and tempered action of anormal j -group B on a Fr´echet algebra A and µ , µ be two families of temperedweights with sub-multiplicative degree ( L , R ) , ( L , R ) . For g ∈ B , the map ˆ α g : F (cid:2) g ∈ B α g (cid:0) F ( g ) (cid:1)(cid:3) , is continuous on B µ i ( B , A ∞ ) , i = 1 , . Moreover, given ( θ, ~τ ) ∈ R ∗ × Θ N , ˆ α definesan action of B by automorphisms of the deformed product ⋆ θ,~τ , in the sense thatfor all F ∈ B µ ( B , A ∞ ) and F ∈ B µ ( B , A ∞ ) , we have for all g ∈ B : ˆ α g (cid:0) F ⋆ θ,~τ F ′ (cid:1) = ˆ α g ( F ) ⋆ θ,~τ ˆ α g ( F ′ ) in B ν ( B , A ∞ ) , with ν = { µ L ,j,k ,j,k µ ,j,kL ,j,k } j,k ∈ N . Proof.
For F ∈ B µ ( B , A ∞ ), X, Y ∈ U ( b ) and g, g ′ ∈ B , we have Y α (cid:0) e X ˆ α g ( F )( g ′ ) (cid:1) = α g (cid:16) ( Ad g − Y ) α (cid:0) e XF ( g ′ ) (cid:1)(cid:17) . This entails that k ˆ α g ( F ) k ( j,k ) ,k ′ ,µ = sup X ∈ U k ′ ( b ) sup g ′ ∈ B sup Y ∈ U k ( b ) k Y α (cid:0) e X ˆ α g ( F )( g ′ ) (cid:1) k j µ j,k ( g ′ ) | X | k ′ | Y | k = sup X ∈ U k ′ ( b ) sup g ′ ∈ B sup Y ∈ U k ( b ) k α g (cid:16) ( Ad g − Y ) α (cid:0) e XF ( g ′ ) (cid:1)(cid:17) k j µ j,k ( g ′ ) | X | k ′ | Y | k ≤ C ( k ) µ αj ( g ) d B ( g ) k sup X ∈ U k ′ ( b ) sup g ′ ∈ B sup Y ∈ U k ( b ) k Y α (cid:0) e XF ( g ′ ) (cid:1) k j µ j,k ( g ′ ) | X | k ′ | Y | k = C ( k ) µ αj ( g ) d B ( g ) k k F k ( j,k ) ,k ′ ,µ , proving the continuity.Next, consider F ∈ B µ ( B , A ∞ ) and F ∈ B µ ( B , A ∞ ), together with ˆ µ andˆ µ , two families of tempered weights that dominate respectively µ and µ . Defining F ,n := F e n ∈ D ( B , A ) and F ,n = F e n ∈ D ( B , A ), with e n ∈ D ( B ) defined in(1.7), from ˆ α g ( F ,n ) = ˆ α g ( F ) e n , ˆ α g ( F ,n ) = ˆ α g ( F ) e n , we deduce from Lemma 1.12 (viii) that { ˆ α g ( F ,n ) } and { ˆ α g ( F ,n ) } converges to { ˆ α g ( F ) } and { ˆ α g ( F ) } in the topologies of B ˆ µ ( B , A ∞ ) and B ˆ µ ( B , A ∞ ) respec-tively. Thus, we can use Lemma 1.44 to get the ˆ α -equivariance at the level ofsmooth compactly supported functions from the commutativity of ˆ α and R ⋆ :ˆ α g (cid:0) F ⋆ θ,~τ F ′ (cid:1) = ˆ α g (cid:0) lim n,n ′ →∞ F n ⋆ θ,~τ F ′ n ′ (cid:1) = lim n,n ′ →∞ ˆ α g (cid:0) F n ⋆ θ,~τ F ′ n ′ (cid:1) = lim n,n ′ →∞ ˆ α g ( F n ) ⋆ θ,~τ ˆ α g ( F ′ n ′ ) = ˆ α g ( F ) ⋆ θ,~τ ˆ α g ( F ′ ) , .1. THE DEFORMED PRODUCT 69 in B ν ( B , A ∞ ), with ν = { µ L ,j,k ,j,k µ ,j,kL ,j,k } . (cid:3) We are now prepared to state the main result of the first part of this memoir:
Theorem . Let ( A , α ) be a Fr´echet algebra endowed with a tempered and strongly continuous action of anormal j -group B . Let also θ ∈ R ∗ and ~τ ∈ Θ N . Then, ( A ∞ , ⋆ αθ,~τ ) is an associativeFr´echet algebra with continuous product. Proof.
Let µ α be the family of tempered weights, with sub-multiplicativedegree ( L, R ), associated with the tempered action α as in Definition 4.1. Let a, b ∈ A ∞ , then by Lemma 4.5, α ( a ) , α ( b ) ∈ B µ ( B , A ∞ ), where µ = { µ αj d k B } . Then,since d B is sub-multiplicative of degree (1 , α ( a ) ⋆ θ,~τ α ( b )belongs to B ν ( B , A ∞ ), for ν = { µ α L j j d k B } , and that the map A ∞ × A ∞ → B ν ( B , A ∞ ) , ( a, b ) α ( a ) ⋆ θ,~τ α ( b ) , is continuous. Applying Lemma 4.6 for the Fr´echet algebra A ∞ then yields thatthe composition of maps A ∞ × A ∞ → B ν ( B , A ∞ ) → A ∞ , ( a, b ) α ( a ) ⋆ θ,~τ α ( b ) (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) ( e ) =: a ⋆ αθ,~τ b , is continuous.It remains to prove associativity. With ˆ α defined in Lemma 4.7, we computefor a, b ∈ A ∞ and g ∈ B : α (cid:0) a ⋆ αθ,~τ b (cid:1) ( g ) = α g (cid:0) a ⋆ αθ,~τ b (cid:1) = α g (cid:0) α ( a ) ⋆ θ,~τ α ( b )( e ) (cid:1) = ˆ α g (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) ( e ) . Using Lemma 4.7, we deduce the equality in B ν ( B , A ∞ ) (for the value of ν asindicated above): ˆ α g (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) = ˆ α g ( α ( a )) ⋆ θ,~τ ˆ α g ( α ( b )) . As a short computation shows, for a ∈ A and g ∈ B , we have ˆ α g (cid:0) α ( a ) (cid:1) = L ⋆g − (cid:0) α ( a ) (cid:1) . Thus, using the equivariance of the product ⋆ θ,~τ under the left regularaction, as stated in Theorem 3.9, we get the equalitiesˆ α g ( α ( a )) ⋆ θ,~τ ˆ α g ( α ( b )) = L ⋆g − ( α ( a )) ⋆ θ,~τ L ⋆g − ( α ( b )) = L ∗ g − (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) , in B λ ( B , A ∞ ), for λ = { µ α L j R j j d k B } . Evaluating this equality at the unit element,yields, by Lemma 4.6, the equality in A ∞ (remember that g ∈ B is fixed): α (cid:0) a ⋆ αθ,~τ b (cid:1) ( g ) = L ∗ g − (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) ( e ) = (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) ( g ) . Hence, the functions α (cid:0) a⋆ αθ,~τ b (cid:1) and α ( a ) ⋆ θ,~τ α ( b ) coincide. This implies for a, b, c ∈A ∞ : a ⋆ αθ,~τ (cid:0) b ⋆ αθ,~τ c (cid:1) = (cid:0) α ( a ) ⋆ θ,~τ α (cid:0) b ⋆ αθ,~τ c (cid:1)(cid:1) ( e ) = (cid:0) α ( a ) ⋆ θ,~τ (cid:0) α ( b ) ⋆ θ,~τ α ( c ) (cid:1)(cid:1) ( e ) , and the associativity of ⋆ αθ,~τ on A ∞ follows from associativity of ⋆ θ,~τ on the tripleCartesian product of the space B ν ( B , A ), as stated in Theorem 3.9. (cid:3) Remark . Contrarily to the R d -action case treated in [ ], in the non-Abelian situation the original action is no longer an automorphism of the deformedproduct ⋆ θ,~τ on A ∞ . This can be understood as the chief reason to introduce thewhole oscillatory integrals machinery in chapter 1 and also to consider the spaces B µ ( B , A ). To conclude this section, we establish a formula for the deformed product ⋆ αθ,~τ on A ∞ , which in some sense, is more natural. It will also clarify an importantpoint, namely that the universal deformation of the algebra A = C ru ( B ), for theaction α = R ⋆ coincides with (cid:0) B ( B ) , ⋆ θ,~τ (cid:1) . Proposition . Let ( α, µ α ) be a strongly continuous and tempered actionof a normal j -group B on a Fr´echet algebra A . Then, for a, b ∈ A ∞ and θ ∈ R ∗ , ~τ ∈ Θ N , we have (4.3) a ⋆ αθ,~τ b = ^ Z B × B K θ,~τ (cid:0) α ( a ) ⊗ α ( b ) (cid:1) , where we denote α ( a ) ⊗ α ( b ) : B × B → A ∞ : ( x, y ) α x ( a ) α y ( b ) . Proof.
Since for a ∈ A ∞ , the element α ( a ) belongs to B µ ( B , A ∞ ), µ = { µ αj d k B } , by Lemma 4.5 and the Leibniz rule, we get that α ( a ) ⊗ α ( b ) ∈ B µ ⊗ µ ( B × B , A ∞ ) , which shows that the right hand side of (4.3) is indeed well defined. Next, byconstruction we have α ( a ) ⋆ θ,~τ α ( b ) = ^ Z B × B K θ,~τ (cid:16) R ⊗ R (cid:0) α ( a ) , α ( b ) (cid:1)(cid:17) ∈ B λ ( B , A ∞ ) , with λ = { µ α L j R j j d k B } , where the map R⊗R has been defined in Lemma 1.42. Now, using Lemma 1.44, weget with the element e n ∈ D ( B ) defined in (1.7), n ∈ N , the equality in B λ ( B , A ∞ ): α ( a ) ⋆ θ,~τ α ( b ) = lim n,m →∞ Z B × B K θ,~τ ( x, y ) R ⋆x (cid:0) e n α ( a ) (cid:1) R ⋆y (cid:0) e m α ( b ) (cid:1) d B ( x ) d B ( y ) . By Lemma 4.6, we know that the evaluation at the neutral element is continuousfrom B λ ( B , A ∞ ) to A ∞ . Thus we get a ⋆ αθ,~τ b = (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) ( e )= lim n,m →∞ Z B × B K θ,~τ ( x, y ) e n ( x ) α x ( a ) e m ( y ) α y ( b ) d B ( x ) d B ( y ) , one then concludes using Proposition 1.32. (cid:3) Corollary . Let θ ∈ R ∗ and ~τ ∈ Θ N . For A = C ru ( B ) and α = R ⋆ , wehave (cid:0) A ∞ , ⋆ αθ,~τ (cid:1) = (cid:0) B ( B ) , ⋆ θ,~τ (cid:1) . Proof.
By Lemma 1.8 (ii), the set of smooth vectors in C ru ( B ) for the right-regular action is B ( B ). By the Proposition above, their algebraic structures coincidetoo. (cid:3) Recall that C ru ( B ) denotes the C ∗ -algebra of right uniformly continuous and bounded func-tions on B , endowed with the sup-norm. .2. RELATION WITH THE FIXED POINT ALGEBRA 71 Under slightly more restrictive conditions on the tempered action ( α, µ α ), weexhibit a relationship between the deformed Fr´echet algebra ( A ∞ , ⋆ αθ,~τ ) and a fixedpoint subalgebra of (cid:0) B µ α ( B , A ) , ⋆ θ,~τ (cid:1) . This construction is very similar to theconstruction of isospectral deformations of Connes and Dubois-Violette [ ]. So,throughout this paragraph, we still assume that A is a Fr´echet algebra carrying astrongly continuous and tempered action α of a normal j -group B . But now, wefurther assume that the action is almost-isometric . By this, we mean that thereexists a family of tempered weights µ α such that for a ∈ A and all g ∈ B , we have k α g ( a ) k j = µ αj ( g ) k a k j . Example . For any Lie group G , take A = L p ( G ), p ∈ [1 , ∞ ). Then, onthis Banach space (not algebra), the right regular action is almost isometric withassociated weight given by ∆ /pG .Also, to simplify the discussion below, we assume that each weight µ αj is sub-multiplicative. We start with the simple observation that for any family of temperedweights µ , the extended action (defined in Lemma 4.7) ˆ α on B µ ( B , A ), commuteswith the left regular action L ⋆ . This leads us to defined the commuting compositeaction β := ˆ α ◦ L ⋆ = L ⋆ ◦ ˆ α , explicitly given by: (cid:0) β g F (cid:1) ( g ) := α g (cid:0) F ( g − g ) (cid:1) , g, g ∈ B , F ∈ B µ ( B , A ) . Note also that by Lemma 1.12 (ii) and Lemma 4.7, for fixed g ∈ B , β g sendscontinuously B µ ( B , A ) to B ν ( B , A ), with ν = { µ R j j } . Thus, in present context ofsub-multiplicative weights, β g is continuous on B µ ( B , A ). Now observe that foran almost isometric action of a Lie group G on a Fr´echet algebra A , the map α : a [ g α g ( a )], is an isometric embedding of A ∞ into B µ α ( G, A ). Indeed forall j, k ∈ N , we have k α ( a ) k j,k,µ α = sup X ∈ U k ( g ) sup g ∈ G k (cid:0) e X α ( a ) (cid:1) ( g ) k j µ αj ( g ) | X | k = sup X ∈ U k ( g ) sup g ∈ G k α g (cid:0) X α a (cid:1) k j µ αj ( g ) | X | k = sup X ∈ U k ( g ) k X α a k j | X k | = k a k j,k . (4.4)By (cid:0) B µ α ( B , A ) (cid:1) β , we denote the closed subspace of B µ α ( B , A ) of fixed points forthe action β . It is then immediate to see that the image of A ∞ under α liesinside (cid:0) B µ α ( B , A ) (cid:1) β . Reciprocally, an element F of (cid:0) B µ α ( B , A ) (cid:1) β , satisfies F ( g ) = α g (cid:0) F ( e ) (cid:1) for all g ∈ B , i.e . F = α ( a ) with a := F ( e ) ∈ A . But by our assumptionof almost-isometry and (4.4), we have k a k j,k = k F k j,k,µ α , for all j, k ∈ N , andthus a = F ( e ) has to be smooth. This proves that α : A ∞ → (cid:0) B µ α ( B , A ) (cid:1) β is anisomorphism of Fr´echet spaces, which is isometric for each seminorm. Moreover,the map α is an algebra homomorphism. Indeed, by the arguments given in theproof of Theorem 4.8, applied to the case of an almost-isometric action with sub-multiplicative weights µ α , for all a, b ∈ A ∞ we have the equality α (cid:0) a ⋆ αθ,~τ b (cid:1) = α ( a ) ⋆ θ,~τ α ( b ) in (cid:0) B µ α ( B , A ) (cid:1) β , which also shows that (cid:0) B µ α ( B , A ) (cid:1) β is an algebra for ⋆ θ,~τ . In summary, we haveproved the following: Proposition . Let θ ∈ R ∗ , ~τ ∈ Θ N and let ( A , α, µ α , B ) be a Fr´echetalgebra endowed with a strongly continuous tempered and almost-isometric action ofa normal j -group B . Then, we have an isometric isomorphism of Fr´echet algebras: (cid:0) A ∞ , ⋆ αθ,~τ (cid:1) ≃ (cid:16)(cid:0) B µ α ( B , A ) (cid:1) β , ⋆ θ,~τ (cid:17) . Remark . We stress that the assumption of sub-multiplicativity for thefamily of weights µ α , associated with the tempered action α , is in fact irrelevant inthe previous result. However it is unclear to us whether a similar statement holdswithout the assumption of almost-isometry. To conclude with the deformation theory at the level of Fr´echet algebras, weestablish some functorial properties. We come back to the general setting of astrongly continuous and tempered action ( α, µ α ) of a normal j -group B on a Fr´echetalgebra A (i.e . we no longer assume that the action is almost isometric). We startwith the question of algebra homomorphisms. Proposition . Let ( A , {k . k j } , α ) , ( F , {k . k ′ j } , β ) be two Fr´echet algebrasendowed with strongly continuous and tempered actions of a normal j -group B byautomorphisms. Let also T : A → F be a continuous homomorphism which inter-twines the actions α and β . Then for any θ ∈ R ∗ and ~τ ∈ Θ N , the map T restrictsto a homomorphism from ( A ∞ , ⋆ αθ,~τ ) to ( F ∞ , ⋆ βθ,~τ ) . Proof.
Since by assumption T ◦ α = β ◦ T , we get for any P ∈ U ( b ) that T ◦ P α = P β ◦ T , which entails that T restricts to a continuous map from A ∞ to F ∞ . The remaining part of the statement follows then by Lemma 1.37. (cid:3) Next, we prove that if a Fr´echet algebra is endowed with a continuous invo-lution, then the latter will also define a continuous involution for the deformedproduct, under the mild condition that ¯ τ ( − a ) = τ ( a ). Indeed, the latter impliesthat K θ,τ ( x , x ) = K θ,τ ( x , x ) , so by Lemma 1.44, we get: Proposition . Let ( A , α ) be a Fr´echet algebras endowed with a stronglycontinuous tempered action of a normal j -group B . Assuming that for θ ∈ R ∗ and ~τ ∈ Θ N , we have ¯ τ j ( − a ) = τ j ( a ) , j = 1 , . . . , N , then any continuous involution of A ∞ is a continuous involution of ( A ∞ , ⋆ αθ,~τ ) too. In a similar way, we deduce from Lemma 1.44 that the deformation is idealpreserving:
Proposition . Let ( A , α ) be a Fr´echet algebras endowed with a stronglycontinuous tempered action of a normal j -group B and θ ∈ R ∗ , ~τ ∈ Θ N . If I is aclosed α -invariant ideal of A , then I ∞ is a closed ideal of ( A ∞ , ⋆ αθ,~τ ) . We now examine the consequence of the fact that the constant function is theunit of (cid:0) B ( B ) , ⋆ θ,~τ (cid:1) . In Lemma 7.22 we will see how to suppress this extra condition. .3. FUNCTORIAL PROPERTIES OF THE DEFORMED PRODUCT 73
Proposition . Let ( A , α ) be a Fr´echet algebras endowed with a stronglycontinuous and tempered action of a normal j -group B and θ ∈ R ∗ , ~τ ∈ Θ N . If a ∈ A ∞ is fixed by the action α , then for b ∈ A ∞ , we have a ⋆ αθ,~τ b = ab, b ⋆ αθ,~τ a = ba . Proof.
This is a consequence of Proposition 3.11 together with the definingrelation of the deformed product: a ⋆ αθ,~τ b = (cid:0) α ( a ) ⋆ θ,~τ α ( b ) (cid:1) ( e ) = (cid:0) a ⋆ θ,~τ α ( b ) (cid:1) ( e ) = (cid:0) aα ( b ) (cid:1) ( e ) = ab . The second equality is entirely similar. (cid:3)
Next, we study the question of the existence of a bounded approximate unit forthe Fr´echet algebra ( A ∞ , ⋆ αθ,~τ ). We recall that a Fr´echet algebra ( A , {k . k j } ) admitsa bounded approximate unit if there exists a net { e λ } λ ∈ Λ of elements of A suchthat for any a ∈ A , the nets { ae λ } λ ∈ Λ and { e λ a } λ ∈ Λ converges to a and such thatfor each j ∈ N , there exists C j > λ ∈ Λ, we have k e λ k j ≤ C j . Proposition . Let ( A , α ) be a Fr´echet algebra endowed with a stronglycontinuous and tempered action of a normal j -group B and such that A admits abounded approximate unit. Then for any θ ∈ R ∗ , ~τ ∈ Θ N , the Fr´echet algebra ( A ∞ , ⋆ αθ,~τ ) admits a bounded approximate unit too. Proof.
Let { f λ } be a net of bounded approximate units for A , let also 0 ≤ ψ ∈ D ( B ) be of L -norm one and define e λ := Z B ψ ( g ) α g ( f λ ) d B ( g ) . Observe that even if { f λ } is not smooth, { e λ } is. Indeed, for all X ∈ U ( b ), we have X α e λ = Z B Xψ ( g ) α g ( f λ ) d B ( g ) , and we get for the semi-norms defining the topology of A ∞ , with µ α the family oftempered weights associated to the temperedness of the action α : k e λ k j,k = sup X ∈ U k ( b ) k X α e λ k j | X | k ≤ sup X ∈ U k ( b ) Z B | Xψ | ( g ) k α g ( f λ ) k j d B ( g ) ≤ sup X ∈ U k ( b ) Z B | Xψ | ( g ) | X | k µ αj ( g ) d B ( g ) × k f λ k j . Hence, the net { e λ } belongs to A ∞ and is semi-norm-wise bounded in λ ∈ Λ as k f λ k j is. Next, we show that it is indeed an approximate unit for A ∞ : Since R ψ = 1, we first note that for any a ∈ A e λ a − a = Z B ψ ( g ) (cid:0) α g ( f λ ) a − a (cid:1) d B ( g ) = Z B ψ ( g ) α g (cid:0) f λ α g − ( a ) − α g − ( a ) (cid:1) d B ( g ) , which gives k e λ a − a k j ≤ Z B ψ ( g ) µ αj ( g ) k f λ α g − ( a ) − α g − ( a ) k j d B ( g ) , which converges to zero because k f λ α g − ( a ) − α g − ( a ) k j does by assumptions andbecause ψ is compactly supported. The general case is treated recursively exactlyas in the proof of Lemma 1.8 (viii). Hence, A ∞ (with its original algebraic struc-ture) admits a bounded approximate unit too. Now, we will prove that a bounded approximate unit for A ∞ is also a bounded approximate unit for ( A ∞ , ⋆ αθ,~τ ). So,let { e λ } be any bounded approximate unit for A ∞ . First observe that if we viewthe product ⋆ θ,~τ as a bilinear map ⋆ θ,~τ : B ( B ) × B µ ( B , A ∞ ) → B ν ( B , A ∞ ) , µ = { µ αj d k B } j,k ∈ N , ν = { µ αL j j d k B } j,k ∈ N , a slight adaptation of the arguments of Proposition 3.11 shows that for all a ∈ A ∞ :1 ⋆ θ,~τ α ( a ) = α ( a ) , where 1 denotes the unit element of B ( B ). Combining this with Proposition 4.10gives the equality in A ∞ : e λ ⋆ αθ,~τ a − a = ^ Z B × B K θ,~τ (cid:0) α ( e λ ) ⊗ α ( a ) − ⊗ α ( a ) (cid:1) , where α ( e λ ) ⊗ α ( a ) − ⊗ α ( a ):= (cid:2) ( x, y ) ∈ B × B α x ( e λ ) α y ( a ) − α y ( a ) (cid:3) ∈ B µ ⊗ µ ( B × B , A ∞ ) . But by Proposition 1.36, we know that given ( j, k ) ∈ N , there exist positive integers ~r ∈ N N , such that (where the differential operator D ~r is given in (1.22)), we have e λ ⋆ αθ,~τ a − a = Z B × B K θ,~τ ( x, y ) D ~r (cid:0) α x ( e λ ) α y ( a ) − α y ( a ) (cid:1) d B ( x ) d B ( y ) , with the integral being absolutely convergent for the semi-norm k . k j,k of A ∞ . Now,take an increasing sequence { C n } n ∈ N of relatively compact open subsets in G , suchthat lim n C n = G and fix ε >
0. By the absolute convergence in the semi-norm k . k j,k of the integral above and since the net { e λ } λ ∈ Λ is bounded in the semi-norm k . k j,k , there exists n ∈ N such that (cid:13)(cid:13)(cid:13) Z B × B \ C n K θ,~τ ( x, y ) D ~r (cid:0) α x ( e λ ) α y ( a ) − α y ( a ) (cid:1) , d B ( x ) d B ( y ) (cid:13)(cid:13)(cid:13) j,k ≤ ε . Moreover, since { e λ } is an approximate unit for A ∞ , from a compactness argument,we deduce that for any n ∈ N , we havelim λ (cid:13)(cid:13)(cid:13) Z C n K θ,~τ ( x, y ) D ~r (cid:0) α x ( e λ ) α y ( a ) − α y ( a ) (cid:1) d B ( x ) d B ( y ) (cid:13)(cid:13)(cid:13) j,k = 0 . This concludes the proof as the arguments for a ⋆ αθ,~τ e λ are similar. (cid:3) Lastly, we show that the deformation associated with a normal j -group coincideswith the iterated deformations of each of its elementary normal j -subgroups. Proposition . Let B be a normal j -group with Pyatetskii-Shapiro decom-position B = B ′ ⋉ S , where B ′ is a normal j -group and S is an elementary normal j -group. Let A be a Fr´echet algebra endowed with a strongly continuous and tem-pered action ( α, µ α ) of B . Denote by α B ′ (respectively by α S ) the restriction of α to B ′ (respectively to S ). For C a subspace of A , denote by C ∞ B (respectively by C ∞ B ′ , C ∞ S ) the set of smooth vectors in C for the action of B (respectively of B ′ , S ). Then,for θ ∈ R ∗ and ~τ = ( ~τ ′ , τ ) ∈ Θ N +1 ( N is the number of elementary factors in B ′ ),we have (cid:0) ( A ∞ S , ⋆ α S θ,τ ) ∞ B ′ , ⋆ α B ′ θ,~τ ′ (cid:1) = ( A ∞ B , ⋆ αθ,~τ ) . .3. FUNCTORIAL PROPERTIES OF THE DEFORMED PRODUCT 75 Proof.
Observe that being the restrictions of a strongly continuous and tem-pered action, the action α S of S on A is also strongly continuous and tempered. Butthe action α B ′ of B ′ on A ∞ S is also strongly continuous (which is rather obvious)and tempered. To see that, note that for g ′ ∈ B ′ and a ∈ A ∞ S , we have k α B ′ g ′ ( a ) k j,k = sup X ∈ U k ( s ) k X α S α B ′ g ′ ( a ) k j | X | k = sup X ∈ U k ( s ) k α B ′ g ′ (cid:0) ( Ad g ′− X ) α S a (cid:1) k j | X | k ≤ µ αj ( g ′ ) sup X ∈ U k ( s ) k ( Ad g ′− X ) α S a k j | X | k . As B ′ acts on S by conjugation, it acts on U k ( s ) and by Lemma 1.14, we deducethat(4.5) k α B ′ g ′ ( a ) k j,k ≤ C ( k ) µ αj ( g ′ ) d B ( g ′ ) k k a k j,k . By Lemma 1.5, we have d B (cid:12)(cid:12) B ′ ≍ d B ′ + (cid:2) g ′ (1 + | R g ′ | + | R g ′− | ) (cid:3) , and by Lemma 2.34, the extension homomorphism R is tempered. Hence, theaction α B ′ of B ′ on A ∞ S is tempered with associated family of tempered weightsgiven by { µ αj d k B (cid:12)(cid:12) B ′ } ( j,k ) ∈ N . Note also that the subspace of smooth vectors for B coincides with the subspace of smooth vectors for B ′ within the subspace of smoothvectors for S , i.e . A ∞ B = ( A ∞ S ) ∞ B ′ . Indeed, the inclusion A ∞ B ⊂ ( A ∞ S ) ∞ B ′ is clear since a ∈ ( A ∞ S ) ∞ B ′ if and only if for all X ′ ∈ U ( b ′ ), all X ∈ U ( s ) and all j ∈ N , we have k X ′ α B ′ X α S a k j < ∞ , and X ′ X ∈ U ( b ). But this also gives the reversed inclusion since [ b ′ , s ] ⊂ s , anyelement of U ( b ) can be written as a finite sum of elements of the form X ′ X , with X ′ ∈ U ( b ′ ) and X ∈ U ( s ).Next, we show that the action α B ′ of B ′ is by automorphisms on the deformedFr´echet algebra ( A ∞ S , ⋆ α S θ,τ ). First, by Proposition 4.10 and Lemma 1.44, we getwith the elements e n ∈ D ( S ) defined in (1.7), n ∈ N , and for a, b ∈ A ∞ S : a ⋆ α S θ,τ b = ^ Z S × S K θ,τ (cid:0) α ( a ) ⊗ α ( b ) (cid:1) = lim n,m →∞ Z S × S K θ,τ ( x, y ) e n ( x ) α S x ( a ) e m ( y ) α S y ( b ) d S ( x ) d S ( y ) . Observe also that (4.5) shows that for g ′ ∈ B ′ fixed, the operator α B ′ g ′ is continuouson A ∞ S . From this and the absolute convergence of the integrals in the product ⋆ α S θ,τ at the level of compactly supported functions, we deduce that for a, b ∈ A ∞ S and g ′ ∈ B ′ : α B ′ g ′ (cid:0) a ⋆ α S θ,τ b (cid:1) = lim n,m →∞ α B ′ g ′ (cid:16) Z S × S K θ,τ ( x, y ) e n ( x ) α S x ( a ) e m ( y ) α S y ( b ) d S ( x ) d S ( y ) (cid:17) = lim n,m →∞ Z S × S K θ,τ ( x, y ) e n ( x ) α g ′ x ( a ) e m ( y ) α g ′ y ( b ) d S ( x ) d S ( y )= lim n,m →∞ Z S × S K θ,τ ( x, y ) e n ( x ) α S R g ′ ( x ) (cid:0) α B ′ g ′ ( a ) (cid:1) e m ( y ) α S R g ′ ( y ) (cid:0) α B ′ g ′ ( b ) (cid:1) d S ( x ) d S ( y ) . Remember that R ∈ Hom (cid:0) B ′ , Aut ( S , s, ω S ) ∩ Sp(
V, ω ) (cid:1) , where ( V, ω ) is the symplectic vector space attached to S . Using the invariance ofthe two-point kernel and of the left Haar measure under the action of Sp( V, ω ) = Aut ( S ) ∩ Aut ( S , s, ω S ), we get: α B ′ g ′ (cid:0) a ⋆ α S θ,τ b (cid:1) = lim n,m →∞ Z S × S K θ,τ ( x, y ) e n ( x ) α S x (cid:0) α B ′ g ′ ( a ) (cid:1) e m ( y ) α Sy (cid:0) α B ′ g ′ ( b ) (cid:1) d S ( x ) d S ( y )= α B ′ g ′ ( a ) ⋆ α S θ,τ α B ′ g ′ ( b ) . Thus, both Fr´echet algebras (cid:0) ( A ∞ S , ⋆ α S θ,τ ) ∞ B ′ , ⋆ α B ′ θ,~τ ′ (cid:1) and ( A ∞ B , ⋆ αθ,~τ ) are well definedand their underlying sets coincide. It remains to show that their algebraic struc-tures coincide too. But this follows from Proposition 1.39 as the extension homo-morphism R of B = B ′ ⋉ R S is tempered. (cid:3) HAPTER 5
Quantization of polarized symplectic symmetricspaces
In the previous chapters, we defined a deformation of every Fr´echet algebra thatadmits a strongly continuous (and tempered) action of a normal j -group. In par-ticular, the method applies to every C ∗ -algebra which the group acts on. However,in that C ∗ -case, our procedure does not yet yield, at this stage, pre- C ∗ -structureson the deformed algebras. To cure this problem (in the case of an isometric action)we will represent our deformed algebras by bounded operators on an Hilbert space.The present chapter consists in defining these Hilbert space representations.The construction relies on defining a unitary representation of the normal j -group B at hand. This unitary representation is obtained as the tensor product ofirreducible unitary representations of the symplectic symmetric spaces (cf . (2.5))underlying the elementary factors S j in the Pyatetskii-Shapiro decomposition B = (cid:0) S N ⋉ . . . (cid:1) ⋉ S of the normal j -group (cf . (2.3)).At the level of an elementary factor S , the unitary representation Hilbert spacewill be defined through a variant of Kirillov’s orbit method when viewing the sym-plectic symmetric space S as a polarized co-adjoint orbit of a central extension of itstransvection group (i.e . the subgroup of the automorphism group of the symplecticsymmetric space, generated by products of even numbers of geodesic symmetries–see Proposition 5.1 below). However, the construction applies to a much moregeneral situation than the one of elementary normal j -groups: the situation ofwhat we call “elementary local symplectic symmetric spaces”. In particular, forthese spaces, we will obtain an explicit formula for the composition of symbols (seeProposition 5.55) which will be our main tool to investigate the problem of C ∗ -deformations in the next sections. We therefore opted, within the present chapter5, to start with presenting this more general situation in the sections 5.1 to 5.6, andthen to later pass to the particular case of elementary normal j -groups in chapter6. Once the elementary case is treated, one then needs to pass to the case of anormal j -group B . In that case, as already mentioned above, the Hilbert space H willconsist in the tensor product of the Hilbert spaces H j representing its elementaryfactors S j . In order to define the B -action on the tensor product Hilbert space H , we will need to represent every elementary group factor S j on each H k forevery k less than or equal to j . The issue here is that the transvection group of S k does not generally contain S j as a subgroup. However, we will show that S j injects into the subgroup of the automorphism group Aut ( S k ) of S k that preservesKirillov’s polarization (see Proposition 6.26) on S k . The latter property will beshown, already in the more general case of local symplectic symmetric spaces, tobe sufficient to extend the action of S k on H k to an action of S j ⋉ S k on H j ⊗ H k
778 5. QUANTIZATION OF POLARIZED SYMPLECTIC SYMMETRIC SPACES (see section 5.7 below). Iterating this result will then lead to our definition of theunitary representation of B on H and its associated symbol calculus.At the level of an elementary normal j -group S , the above mentioned quanti-zation map (allowing to represent our algebras by bounded operators), realizes theprogram of the construction of ‘covariant Moyal quantizers’ in the sense of [ ] (seealso [ , Section 3.5]). Such an object is of the following nature: Let ( M, ω ) be asymplectic manifold and let G be a Lie subgroup of the symplectomorphisms groupSymp( M, ω ), together with ( H , U ), a Hilbert space carrying a unitary representa-tion of G . Then, a covariant Moyal quantizer, is a family { Ω( x ) } x ∈ M of denselydefined self-adjoint operators on H such that U ( g ) Ω( x ) U ( g ) ⋆ = Ω( g.x ) , ∀ ( g, x ) ∈ G × M , (5.1) Tr (cid:0) Ω( x ) (cid:1) = 1 , ∀ x ∈ M , (5.2) Tr (cid:0) Ω( x ) Ω( y ) (cid:1) = δ x ( y ) , ∀ ( x, y ) ∈ M × M , (5.3)where the traces have to be understood in the distributional sense and where δ x ( y )is a shorthand for the reproducing kernel of the Liouville measure on M . Associatedto a Moyal quantizer, there are quantization and dequantization maps, respectivelygiven, on suitable domains and with dµ the Liouville measure on M , byΩ : f Z M f ( x ) Ω( x ) dµ ( x ) , and by σ : A Tr (cid:0) Ω( x ) A (cid:1) . Then, condition (5.1) ensures that both quantization and dequantization maps are G -equivariant. Condition (5.2) is a normalization condition and condition (5.3)says that Ω and σ are formal inverses of each other. At the non-formal level,condition (5.3) together with self-adjointness of the quantizers, implies that Ω isa unitary operator from L ( M ) to L ( H ), the Hilbert space of Hilbert-Schmidtoperators on H . Transporting the algebraic structure from L ( H ) to L ( M ), onetherefore obtains a non-formal and G -equivariant star-product at the level of squareintegrable functions: f ⋆ f := σ (cid:0) Ω( f ) Ω( f ) (cid:1) . This non-formal star-product turns to be a tri-kernel product, with distributionalkernel computable as it is given by the trace of three quantizers: f ⋆ f ( x ) = Z M × M f ( x ) f ( x ) Tr (cid:0) Ω( x ) Ω( x ) Ω( x ) (cid:1) dµ ( x ) dµ ( x ) . To construct such a quantizer for an elementary local symplectic symmetricspaces, we essentially follow the pioneer ideas of Unterberger [ ] and recast themin a much more general geometric context. Note that, however, in the work ofUnterberger the condition (5.3) does not hold but, as we will see in the presentchapter, is restorable by a minor modification of his construction. This conditionis also called the ‘traciality property’ as it eventually allows to prove that: Z M f ⋆ f ( x ) dµ ( x ) = Z M f ( x ) f ( x ) dµ ( x ) , ∀ f , f ∈ L ( M ) . Traciality is a property not shared by every quantization map (for example its failsfor the coherent-state quantization) and proved to be fundamental in the work of .1. POLARIZED SYMPLECTIC SYMMETRIC SPACES 79
Gracia-Bond´ıa and V´arilly. For instance, in [ ], they were able to construct (andto prove uniqueness of) an equivariant quantization map on the (regular) orbits ofthe Poincar´e group in 3 + 1 dimensions solely from the axioms (5.1), (5.2), (5.3). In this section, we introduce a particular class of symplectic symmetric spaces(called hereafter “polarized”) for which we will be able to define a symmetric spacevariant of Kirillov’s orbit method (see the next section). Within the present section,after defining polarized symplectic symmetric spaces, we associate to every suchspace an algebraic object (its “polarization quadruple”) on which we will later baseour unitary representation.Let (
M, s, ω ) be a symplectic symmetric space and let
Aut ( M, s, ω ) be its au-tomorphism group and aut ( M, s, ω ) its derivation algebra (see Definition 2.7 andProposition 2.8). Choose a base point o in M . Then the conjugation by the sym-metry at o yields an involutive automorphism of the automorphism group: σ : Aut ( M, s, ω ) → Aut ( M, s, ω ) , g s o ◦ g ◦ s o =: σ ( g ) . (5.4)The following result is a simple adaptation to the symplectic situation of astandard fact for general symmetric spaces [ ]: Proposition . The smallest subgroup G ( M ) of Aut ( M, s, ω ) that is stableunder σ and that acts transitively on M is a Lie subgroup of Aut ( M, s, ω ) . Itcoincides with group generated by products of an even number of symmetries: G ( M ) = gr { s x ◦ s y | x, y ∈ M } . The group G ( M ) is called the transvection group of M . We now come to the notion of polarized symplectic symmetric spaces:
Definition . A symplectic symmetric space (
M, ω, s ) is said to be polar-izable if it admits a G ( M )-invariant Lagrangian tangent distribution. A choice ofsuch a transvection-invariant distribution ˜ W ⊂ T M determines a polarization of M , in which case one speaks about a polarized symplectic symmetric space.The infinitesimal version of the notion of symplectic symmetric space is givenin the two following definitions (see [ , ]): Definition . A symplectic involutive Lie algebra (shortly a “siLa”)is a triple ( g , σ, ̟ ) where ( g , σ ) is an involutive Lie algebra (shortly an “iLa”)i.e . g is a finite dimensional real Lie algebra and σ is an involutive automorphismof g , and, where ̟ ∈ V g ⋆ is a Chevalley two-cocycle on g (valued in the trivialrepresentation on R ) such that, denoting by g = k ⊕ p , the ± g associated to the involution σ =: id k ⊕ ( − id p ),the cocycle ̟ contains k in its radical and restricts to p × p as a non-degeneratetwo-form. A morphism between two such siLa’s is a Lie algebra homomorphismwhich intertwines both involutions and two-cocycles. Definition . A transvection symplectic triple is a siLa ( g , σ, ̟ ) where Involutions σ are denoted the same way either at the Lie group or Lie algebra level. (i) the action of k on p is faithful, and(ii) [ p , p ] = k . Lemma . Every symplectic symmetric space determines a transvection sym-plectic triple.
Proof.
Let (
M, s, ω ) be a symplectic symmetric space. Define a transvectionsymplectic triple ( g , σ, ̟ ) as follows: g is the Lie algebra of the transvection group G ( M ), σ is the restriction to g of the differential of the involution (5.4) and ̟ = π ⋆⋆e ω o is the pullback of ω at the base point o by the differential π ⋆e : g → T o M ofthe projection π : G ( M ) → M , g g.o . See [ ] for the details. (cid:3) Defining the notion of isomorphism in the obvious way, one knows from [ , ]the following result. It is a symplectic adaptation of the classical analogue forgeneral affine symmetric spaces [ ]. Proposition . The correspondence ( M, s, ω ) ( g , σ, ̟ ) described aboveinduces a bijection between the isomorphism classes of simply connected symplecticsymmetric spaces and the isomorphism classes of transvection symplectic triples. In the above statement, the “reverse direction” (i.e . ( g , σ, ̟ ) ( M, s, ω )) isobtained as follows. One starts by considering the (abstract) connected simplyconnected Lie group G whose Lie algebra is g . Then the automorphism σ of g uniquely determines an automorphism of G , again denoted by σ . The connectedcomponent K of the subgroup of G constituted by the elements that are fixed by σ is then automatically a closed subgroup of G . The coset space M := G/K is thennaturally a connected smooth manifold which is also simply connected consequentlyto the connectedness of K and simple connectedness of G . One then check that theformulae(5.5) s gK ( g ′ K ) := gσ ( g − g ′ ) K , define a structure of symmetric space (
M, s ) in the sense of Loos. The Lie algebra of K then coincides with k , implying a natural isomorphism p → T K ( M ). The latteryields a Lie algebra homomorphism from the original iLa ( g , σ ) to the transvectioniLa of ( M, s ). This homomorphism turns out to be an isomorphism due to the aboveconditions (i) and (ii) imposed on ( g , σ ). Note that from the above exposition, oneextracts Lemma . Every iLa determines a simply connected symmetric space.
Note also that two non-isomorphic iLa’s (when non-transvection) could deter-mine the same symmetric space. More precisely:
Definition . Let ( g j , σ j ) ( j = 1 ,
2) be two iLa’s with associated simplyconnected symmetric spaces denoted by ( M j , s j ) respectively. One says that they determine the same simply connected symmetric space if M and M areisomorphic as symmetric spaces.Of course, one has an analogous definition in the symplectic case.Now given a siLa ( g , σ, ̟ ), the two-cocycle ̟ is generally not exact, or equiva-lently, the symplectic action of the transvection group on the symplectic symmetricspace is not Hamiltonian (see [ , ]). However, it is always possible to centrallyextend the transvection group in such a way that the extended group acts on M .1. POLARIZED SYMPLECTIC SYMMETRIC SPACES 81 in a Hamiltonian way. The associated moment mapping then is a symplectic equi-variant covering onto a co-adjoint orbit, in accordance with the classical generalresult for Hamiltonian homogeneous symplectic spaces [ ]. The situation we con-sider in the present article concerns such non-exact transvection triples underlyingpolarized symplectic symmetric spaces. Lemma . Let ( M, s, ω ) be a symplectic symmetric space, polarized by atransvection-invariant Lagrangian distribution ˜ W ⊂ T M . These data correspond(via the correspondence of Proposition 5.6) to a k -invariant Lagrangian subspace W in p . Proof.
Under the linear isomorphism π ⋆e | p : p → T o M the subspace ˜ W o of T o M corresponds to a Lagrangian subspace W of the symplectic vector space( p , ̟ ). (cid:3) According to the previous Lemma, we use the following terminology:
Definition . A siLa ( g , σ, ̟ ) is called polarized if it is endowed with W ,a k -invariant Lagrangian subspace of ( p , ̟ ).Let ( g , σ, ̟ ) be a non-exact transvection symplectic triple (i.e . the Chevalleytwo-cocycle ̟ is not exact) polarized by a Lagrangian subspace W ⊂ p . Let usconsider D , the algebra of W -preserving symplectic endomorphisms of p . Notethat the faithfulness condition (i) of Definition 5.4 implies the inclusion k ⊂ D .The vector space D ⊕ p then naturally carries a structure of Lie algebra (containing g ) that underlies a siLa. We centrally extend the latter in order to define a newsiLa: L := D ⊕ p ⊕ R Z , with table given by[
X, Y ] L := [ X, Y ] + ̟ ( X, Y ) Z , [ X, Z ] L := 0 , ∀ X, Y ∈ D ⊕ p , where [ ., . ] denotes the Lie bracket in D ⊕ p and where we have extended the 2-form ̟ on p to a 2-form on the entire Lie algebra D ⊕ p by zero on D . Lemma . Let ( g , σ, ̟ ) be a non-exact polarized transvection symplectictriple. Within the notations given above, consider the element ξ ∈ L ⋆ defined by h ξ, Z i = 1 , ξ (cid:12)(cid:12) D ⊕ p = 0 . Define moreover: ˜ D := D ⊕ R Z , σ L := id ˜ D ⊕ ( − id p ) , and ˜ g = g ⊕ R Z , ˜ k = k ⊕ R Z , ˜ σ := id ˜ k ⊕ ( − id p ) . Then, the triples ( L , σ L , δξ ) and (˜ g , ˜ σ, δξ (cid:12)(cid:12) ˜ g ) are exact siLa’s. Proof.
We give the proof for the first triple ( L , σ L , δξ ) only, the second casebeing handled in a similar way. Since for X, Y ∈ p , we have [ X, Y ] ∈ k ⊂ D , we get δξ ( X, Y ) = h ξ, [ X, Y ] L i = h ξ, [ X, Y ] + ̟ ( X, Y ) Z i = ̟ ( X, Y ) , which at once proves closedness, non-degeneracy and ˜ D -invariance of the 2-form δξ on p . (cid:3) Remark . The exact siLa’s ( L , σ L , δξ ) and (˜ g , ˜ σ, δξ (cid:12)(cid:12) ˜ g ) and the non-exactsiLa ( g , σ, ̟ ) all three determine the same simply connected symplectic symmetricspace. Definition . Given an exact siLa ( g , σ, ̟ ) (i.e . ̟ = δξ for an element ξ ∈ g ⋆ ), by a polarization affiliated to ξ , we mean a σ -stable Lie subalgebra b of g containing k and maximal for the property of being isotropic with respect to thetwo-form ̟ .The following statement is classical (see e.g . [ ]). Lemma . Let ( G, σ ) be a connected involutive Lie group i.e . a connectedLie group G equipped with an involutive automorphism σ . Let us denote by K theconnected component of the subgroup of G constituted by the σ -fixed elements. Then K must be closed and Formula (5.5) defines a structure of symmetric space on thequotient manifold M = G/K . Note that in this slightly more general situation, M need not necessarily besimply connected.Within the above setting let us denote by ( g , σ ) the involutive Lie algebraassociated to ( G, σ ). Let us furthermore assume that it underlies an exact siLa withpolarization b as in Definition 5.13. In that context, M automatically becomes apolarized symplectic symmetric space. Denote by B := exp { b } the analytic (i.e . connected) Lie subgroup of G with Lie algebra b . One has K ⊂ B and we willalways assume B to be closed in G . Since B is stable under σ , the coset space G/B admits the following natural family of involutions σ :(5.6) M × G/B → G/B , ( gK, g B ) σ gK ( g B ) := gσ ( g − g ) B .
Definition . With the same notations as above, the quadruple (
G, σ, ξ, B )is called a polarization quadruple . Its infinitesimal version ( g , σ, ξ, b ) is calledthe associated infinitesimal polarization quadruple . A morphism betweentwo polarization quadruples ( G j , σ j , ξ j , B j ), j = 1 ,
2, is defined as a Lie grouphomomorphism φ : G → G , that intertwines the involutions, such that φ ( B ) ⊂ B and such that, denotingagain by φ its differential at the unit element, one has φ ⋆ ξ = ξ .The map (5.6) corresponds to an ‘action’ of the symmetric space M = G/K on the manifold
G/B . The following result is a consequence of immediate compu-tations.
Lemma . Let ( G, σ, ξ, B ) be a polarization quadruple. Then, the followingproperties hold for all x and y in M : σ x = Id G/B , σ x ◦ σ y ◦ σ x = σ s ( x,y ) . Moreover, we have the G -equivariance property: g ◦ σ x ◦ g − = σ g.x , ∀ g ∈ G , ∀ x ∈ M .
We also observe that under mild conditions on the modular functions of G and B , a G -invariant and M -invariant measure always exists on the manifold G/B : .1. POLARIZED SYMPLECTIC SYMMETRIC SPACES 83 Lemma . Let ( G, σ ) be an involutive Lie group and B a σ -stable closedsubgroup of G such that the modular function of B coincides with the restriction to B of the modular function of G . Then, there exists a (unique up to normalization)Borelian measure d G/B on the manifold
G/B which is both invariant under G andunder the action of M = G/K given in (5.6) . Proof.
Under the closedness condition of B and under the coincidence as-sumption for the modular functions, it is well known that there exists a (uniqueup to normalization) G -invariant Borelian measure d oG/B on G/B . Now, defined
G/B := d oG/B + σ ⋆K d oG/B . As σ K is an involution, the latter measure is σ K -invariant. Moreover, from L ⋆g ◦ σ ⋆K = σ ⋆K ◦ L ⋆σ ( g ) on G/B for all g ∈ G , we deducethat d G/B is also G -invariant. By uniqueness of d oG/B , the latter is a multiple ofthe former. To conclude with the M -invariance, it suffices to observe that for all g ∈ G , we have σ ⋆gK = L ⋆g ◦ σ ⋆K ◦ L ⋆g − . (cid:3) We end this section by constructing two canonical exact polarization quadruplesout of a non-exact transvection triple. We omit the proof which is immediate.
Proposition . Let ( g , σ, ̟ ) be a non-exact transvection symplectic triplepolarized by W ⊂ p . Within the context of Lemma 5.11, we set B := ˜ D ⊕ W and b := ˜ k ⊕ W . Then the quadruples ( L , σ L , ξ, B ) and (˜ g , ˜ σ, ξ (cid:12)(cid:12) ˜ g , b ) are polarizationquadruples. The latter observation leads us to introduce the following terminology:
Definition . The polarization quadruple ( L , σ L , ξ, B ) associated to a non-exact polarized transvection triple ( g , σ, ̟ ), is called the (infinitesimal) full po-larization quadruple . The sub-quadruple (˜ g , ˜ σ, ξ (cid:12)(cid:12) ˜ g , b ) is called the associated(infinitesimal) transvection quadruple .In the sequel, we will denote by L the connected, simply connected Lie groupwith Lie algebra L and we will consider the connected Lie subgroup ˜ G of L tangentto ˜ g . We will denote by B (respectively B ) the connected Lie subgroup of L (respectively of ˜ G ) associated to B (respectively to b ).We summarize the present section by the following Proposition . (i) Every symplectic symmetric space uniquely de-termines a transvection symplectic triple (cf . Definition 5.4). (ii)
Every siLa (cf . Definition 5.3) uniquely determines a simply connectedsymplectic symmetric space. (iii)
In the simply connected and transvection case, the correspondences men-tioned in items (i) and (ii) are inverse to one another. (iv)
Every non-exact transvection symplectic triple ( g , σ, ̟ ) (cf . Definition 5.4with non-exact two-cocycle) uniquely determines a pair of exact polariza-tion quadruples: the associated infinitesimal full polarization quadruple ( L , σ L , ξ, B ) and its transvection sub-quadruple (˜ g , ˜ σ, ξ (cid:12)(cid:12) ˜ g , b ) (cf . Proposi-tion 5.18 and Definition 5.19). (v)
The three siLa’s involved in item (iv): ( g , σ, ̟ ) , (cid:0) ˜ g , ˜ σ, δξ (cid:12)(cid:12) ˜ g (cid:1) and ( L , σ L , δξ ) all determine the same simply connected polarized symplectic symmetricspace (cf . Definition 5.8).
In this section, we fix ( g , σ, ̟ ) a non-exact polarized transvection triple, towhich we associate a (connected and simply connected) transvection quadruple( ˜ G, ˜ σ, ξ, B ), according to the construction underlying Definition 5.19. We startwith the following pre-quantization condition: in the sequel we will always assumethat the character ξ | b : b → R exponentiates to B as a unitary character χ : B → U (1) , b χ ( b ) . By this we mean that we assume the existence of a Lie group homomorphism χ whose differential at the identity coincides with ξ | b . Note that then, the characteris automatically fixed by the restriction to B of the involution: σ ⋆ χ = χ . Of course, the pre-quantization condition is satisfied when the group B is exponen-tial, as it will be the case for Pyatetskii-Shapiro’s elementary normal j -groups:(5.7) χ ( b ) := e i h ξ, log( b ) i , b ∈ B .
Lemma . Let (˜ g , ˜ σ, ξ, b ) be the transvection quadruple of a non-exact trans-vection triple ( g , σ, ̟ ) such that B is exponential. Then, the pre-quantization con-dition is satisfied. Proof.
Since B is exponential, by the BCH formula, the statement will followfrom ξ (cid:0) [ b , b ] ˜ g (cid:1) = 0. By construction of ξ (see Definition 5.11), this will follow if the Z -component of [ b , b ] ˜ g vanishes. But the latter reads ̟ ( b , b ) Z which reduces tozero by Definition 5.13 of a polarization quadruple and by Proposition 5.18 whichshows that (˜ g, ˜ σ, ξ, b ) is indeed a polarization quadruple. (cid:3) We then form the line bundle: E χ := ˜ G × χ C → ˜ G/B , and consider the associated induced representation of ˜ G on the smooth sectionsΓ ∞ ( E χ ). We will denote the latter representation by U χ . Identifying as usualΓ ∞ ( E χ ) with the space of B -equivariant functions:Γ ∞ ( E χ ) ≃ C ∞ ( ˜ G ) B := (cid:8) ˆ ϕ ∈ C ∞ ( ˜ G ) | ˆ ϕ ( gb ) = χ ( b ) ˆ ϕ ( g ) , ∀ b ∈ B, ∀ g ∈ ˜ G (cid:9) , (5.8)the representation U χ is given by the restriction to C ∞ ( ˜ G ) B of the left-regularrepresentation:(5.9) [ U χ ( g ) ˜ ϕ ] ∧ ( g ′ ) := ˆ ϕ ( g − g ′ ) , ∀ ˜ ϕ ∈ Γ ∞ ( E χ ) . We endow the line bundle E χ with the Hermitian structure, defined in terms of theidentification (5.8) by: h gB (cid:0) ˜ ϕ , ˜ ϕ (cid:1) := ˆ ϕ ( g ) ˆ ϕ ( g ) , ∀ ˜ ϕ , ˜ ϕ ∈ Γ ∞ ( E χ ) , gB ∈ ˜ G/B .
We make the assumption that the modular function of B coincides with the restric-tion to B of the modular function of ˜ G . By Lemma 5.17, this condition implies theexistence of a ˜ G -invariant and σ ˜ K -invariant Borelian measure d ˜ G/B on ˜
G/B . Here, σ ˜ K : ˜ G/B → ˜ G/B , gB ˜ σ ( g ) B is the involutive diffeomorphism given in (5.6) forthe involutive pair ( ˜ G, ˜ σ ) and subgroup B , underlying the transvection quadruple .2. UNITARY REPRESENTATIONS OF SYMMETRIC SPACES 85 ( ˜ G, ˜ σ, ξ, B ). We then let H χ be the Hilbert space completion of Γ ∞ c ( E χ ) for theinner product: h ˜ ϕ , ˜ ϕ i := Z ˜ G/B h gB ( ˜ ϕ , ˜ ϕ ) d ˜ G/B ( gB ) . Of course, the induced representation U χ of ˜ G then naturally acts on H χ by unitaryoperators. Now observe that the ˜ σ -invariance of character χ , implies that the pullback under ˜ σ of an equivariant function is again equivariant. Therefore, we get alinear involution: Σ : H χ → H χ , [Σ ˜ ϕ ] ∧ := ˜ σ ⋆ ˆ ϕ . Also, the σ ˜ K -invariance of the measure d ˜ G/B implies: h Σ ˜ ϕ , Σ ˜ ϕ i = Z ˜ G/B h gB (Σ ˜ ϕ , Σ ˜ ϕ ) d ˜ G/B ( gB )= Z ˜ G/B h σ ˜ K ( gB ) ( ˜ ϕ , ˜ ϕ ) d ˜ G/B ( gB ) = h ˜ ϕ , ˜ ϕ i , for all ˜ ϕ , ˜ ϕ ∈ H χ , showing that Σ is not only involutive but also self-adjoint.Thus the element Σ belongs to U sa ( H χ ), the collection of unitary and self-adjointoperators on H χ . When composed with the representation U χ of ˜ G , the operatorΣ satisfies the following properties, whose proofs consist in direct computations: Proposition . Let ( M, s, ω ) be the polarized symplectic symmetric spaceassociated to a transvection quadruple ( ˜ G, ˜ σ, ξ, B ) (cf . Lemmas 5.14 and 5.16).Assume that the modular function of B coincides with the restriction to B of themodular function of ˜ G . Then the map ˜ G → U sa ( H χ ) , g U χ ( g ) Σ U χ ( g ) ∗ , is constant on the left cosets of ˜ K in ˜ G . The corresponding mapping: Ω : M = ˜ G/ ˜ K → U sa ( H χ ) , g ˜ K Ω( g ˜ K ) := U χ ( g ) Σ U χ ( g ) ∗ , defines a unitary representation of the symmetric space M = ˜ G/ ˜ K in the sensethat, for all x , y in M and g in ˜ G , the following representative properties hold: Ω( x ) = Id H χ , Ω( x ) Ω( y ) Ω( x ) = Ω( s x y ) , (5.10) U χ ( g ) Ω( x ) U χ ( g ) ∗ = Ω( g.x ) . Definition . The pair ( H χ , Ω) is called the unitary representation of ( M, s ) induced by the character χ of B .We are now ready to define our prototype of quantization map on a polarizedsymplectic symmetric space: Definition . Let (
M, s, ω ) be a the polarized symplectic symmetric space.Denote by L ( M ) the space of integrable functions on M with respect to the ˜ G -invariant (Liouville) measure d M . Denote by B ( H χ ) the space of bounded linearoperators on the Hilbert space H χ . Consider the ˜ G -equivariant continuous linearmap: Ω : L ( M ) → B ( H χ ) , f Ω( f ) := Z M f ( x ) Ω( x ) d M ( x ) . The latter is called the quantization map of M induced by the transvectionquadruple ( ˜ G, ˜ σ, ξ, B ). Remark . From k Ω( x ) k = k Σ k = 1 (the norm here is the uniform normon B ( H χ )), we get the obvious estimate k Ω( f ) k ≤ k f k , from which the continuityof the quantization map follows. Also, from Proposition 5.22 and from the ˜ G -invariance of d M , the covariance property at the level of the quantization mapreads: U ( g ) Ω( f ) U ( g ) ∗ = Ω( g f ) , ∀ f ∈ L ( M ) , ∀ g ∈ ˜ G , where g f := [ g ˜ K f ( g − g ˜ K )]. The latter equivariance property, under thefull group of automorphisms of the symplectic symmetric space, is an importantdifference between the present “Weyl-type” construction and the classical coherent-state-quantization approach. For instance, as it appears already in the flat case of R n , holomorphic coherent-state-quantization (i.e . Berezin-Toeplitz in that case)yields a equivariance group that is isomorphic to U ( n ) ⋉ C n , while Weyl quantiza-tion is equivariant under the full automorphism group of R n i.e . Sp( n, R ) ⋉ R n .Another essential difference is unitarity (see Proposition 5.48 below). Lastly, thequantization defined above is in general not positive, i.e . for 0 ≤ f ∈ L ( M ), Ω( f )is not necessarily a positive operator. But since Ω( f ) ∗ = Ω( f ), it maps real-valuedfunctions to self-adjoint operators. Remark . The quantization map of Definition 5.24 is a generalization ofthe Weyl quantization, from the point of view of symmetric spaces. Moreover,we will see that for the symmetric space underlying a two-dimensional elementarynormal j -group (i.e . for the affine group of the real line) this construction coincideswith Unterberger’s Fuchs calculus [ ].Our next step is to introduce a functional parameter in the construction of thequantization map. There are several reasons for doing this, among which thereis one of a purely analytical nature: obtaining a quantization map which is aunitary operator from the Hilbert space of square integrable symbols, L ( M ), tothe Hilbert space of Hilbert-Schmidt operators on H χ , denoted by L ( H χ ). Thisunitarity property will enable us to define a non-formal ⋆ -product on L ( M ) in astraightforward way. Definition . Identifying a Borelian function m on ˜ G/B with the operatoron H χ of point-wise multiplication by this function, we letΣ m := m ◦ Σ . When m is locally essentially bounded, the family of operators U χ ( g )Σ m U χ ( g ) ∗ , g ∈ ˜ G , can be defined on the common domain Γ ∞ c ( E χ ). Note however that thelatter family of operators is not necessarily constant on the left cosets of ˜ K in ˜ G and unless σ ⋆ ˜ K m = m − , one loses the involutive property for Σ m (but one alwayskeeps the ˜ G -equivariance). Also, these operators are bounded on H χ , if and only ifthe function m is essentially bounded. But we will see in Theorem 5.43 that in orderto obtain a unitary quantization map we are forced to consider such unboundedΣ m ’s. We mention a simple self-adjointness criterion, interesting on its own. Lemma . Let m be a locally essentially bounded Borelian function on ˜ G/B such that σ ⋆ ˜ K m = m . Define ˜Ω m ( g ) := U χ ( g ) Σ m U χ ( g ) ∗ , g ∈ ˜ G , .2. UNITARY REPRESENTATIONS OF SYMMETRIC SPACES 87 on the domain B g := (cid:8) ϕ ∈ H χ : | m g | ϕ ∈ H χ (cid:9) where m g := [ g B m ( g − g B )] , g ∈ ˜ G .
Then, ˜Ω m ( g ) is self-adjoint on H χ . Moreover, Γ ∞ c ( E χ ) is a common core for all ˜Ω m ( g ) ’s, g ∈ ˜ G . Proof.
Note first that the formal adjoint of Σ m is Σ σ ⋆ ˜ K m . Therefore, when σ ⋆ ˜ K m = m , the operator Ω m ( x ) is symmetric on Γ ∞ c ( E χ ). Next, we remark thatas both Σ and U χ ( g ), g ∈ ˜ G , preserve Γ ∞ c ( E χ ), we get for ϕ ∈ Γ ∞ c ( E χ ):˜Ω m ( g ) ϕ = | m g | ϕ , that is, ˜Ω m ( g ) squares on Γ ∞ c ( E χ ) to a multiplication operator. Since ˜Ω m ( g ) issymmetric on the space of smooth compactly supported sections of E χ , the latterentails that k ˜Ω m ( g ) ϕ k = k| m g | ϕ k , ∀ ϕ ∈ Γ ∞ c ( E χ ) , and thus ˜Ω m ( g ) is well defined on B g . Then, the same computation as above, showsthat ˜Ω m ( g ) is also symmetric on its domain B g . Observe that B g is complete inthe graph norm, given by k ψ k + k| m g | ψ k , and that Γ ∞ c ( E χ ) is dense in B g forthis norm. Thus ˜Ω m ( g ), with domain B g , is a closed operator. Clearly B g ⊂ dom (cid:0) ˜Ω m ( g ) ∗ (cid:1) , since ˜Ω m ( g ) is symmetric on B g .Choose an increasing sequence of relatively compact open sets { C n } n ∈ N in ˜ G/B ,converging to ˜
G/B . For n ∈ N , let χ n be the indicator function of C n . Then ofcourse χ n ϕ ∈ B g for all ϕ ∈ H χ . Note also that for ϕ ∈ B g , we have by definitionof m g and from the relation σ ⋆ ˜ K m = m :˜Ω m ( g ) ϕ = m g Ω( x ) ϕ = Ω( x ) m g ϕ , x = g ˜ K ∈ M .
Thus for g ∈ ˜ G , x = g ˜ K ∈ M , ψ ∈ dom (cid:0) ˜Ω m ( g ) ∗ (cid:1) , ϕ ∈ H χ and using the factthat χ n m g is essentially bounded (i.e . the associated multiplication operator isbounded), we get h ϕ, χ n ˜Ω m ( g ) ∗ ψ i = h ˜Ω m ( g ) χ n ϕ, ψ i = h Ω( x ) m g χ n ϕ, ψ i = h ϕ, χ n m g Ω( x ) ψ i . Using the monotone convergence theorem, we obtain k ˜Ω m ( g ) ∗ ψ k = lim n →∞ k χ n ˜Ω m ( g ) ∗ ψ k = lim n →∞ sup k ϕ k =1 (cid:12)(cid:12) h ϕ, χ n ˜Ω m ( g ) ∗ ψ i (cid:12)(cid:12) = lim n →∞ sup k ϕ k =1 (cid:12)(cid:12) h ϕ, χ n m g Ω( x ) ψ i (cid:12)(cid:12) = lim n →∞ k χ n m g Ω( x ) ψ k = k m g Ω( x ) ψ k = k Ω( x ) m g ψ k = k m g ψ k = k| m g | ψ k , so that necessarily ψ ∈ B g . Thus dom (cid:0) ˜Ω m ( g ) ∗ (cid:1) = B g , as required. Note lastlythat Γ ∞ c ( E χ ) being dense in each B g for the graph norm, it is a common core forall the ˜Ω m ( g ), which are therefore essentially selfadjoint on that domain. (cid:3) Remark . At this early stage of the construction, it is important to ob-serve that our representation of M (Proposition 5.22) and the associated quanti-zation map (Definition 5.24) could have been equally defined starting with the fullpolarization quadruple ( L , σ L , ξ, B ) (see Definition 5.19) of a non-exact polarizedtransvection triple ( g , σ, ̟ ), instead of the transvection quadruple ( ˜ G, ˜ σ, ξ (cid:12)(cid:12) ˜ G , B ). Inparticular, all the results of sections 5.3, 5.4, 5.5 and 5.6 can be thought as arisingfrom the full quadruple. We will make great use of this observation in section 5.7. We next pass to the notion of locality in the context of transvection quadruples,out of which we will be able to give an explicit expression of the operators Ω m ( x )on H χ . Definition . Within the notations of Definition 5.19, we say that thepolarized symplectic symmetric space (
M, s, ω ), associated to a non-exact polarizedtransvection triple ( g , σ, ̟ ), is local whenever there exists a subgroup Q of ˜ G suchthat: (i) The map Q × B → ˜ G , ( q, b ) qb . is a global diffeomorphism. In particular, Q is closed as a subgroup of ˜ G .(ii) For all q ∈ Q and b ∈ B , one has C q (˜ σ ( b ) b − ) ∈ B , where C g ( g ′ ) := gg ′ g − denotes the conjugate action of ˜ G on itself.(iii) For every q ∈ Q , setting ˜ σq =: (˜ σq ) Q (˜ σq ) B relatively to the global de-composition ˜ G = Q.B , one has: χ (cid:0) (˜ σq ) B (cid:1) = 1 . For a local symplectic symmetric space, the identification Q ≃ ˜ G/B allows totransfer the symmetric space structure of the former to the latter:
Lemma . Let ( M, s, ω ) be a local symplectic symmetric space. Then: (i) The mapping: s : Q × Q → Q , ( q, q ′ ) s q ( q ′ ) := q (cid:0) ˜ σ ( q − q ′ ) (cid:1) Q , defines a left-invariant structure of symmetric space on the Lie group Q . (ii) Moreover, the global diffeomorphism Q → ˜ G/B , q qB , intertwines the symmetry s with the involution σ , defined in (5.6) for thetransvection quadruple ( ˜ G, ˜ σ, ξ, B ) : s q q ′ σ q ˜ K ( q ′ B ) , ∀ q, q ′ ∈ Q . (iii)
Under the identification Q ≃ ˜ G/B given above, the ( ˜
G, σ ˜ K ) -invariantmeasure d ˜ G/B on ˜ G/B constructed in Lemma 5.17, becomes a ( ˜
G, s e ) -invariant measure d Q on the Lie group Q , which is also a left-invariantHaar measure on Q . (iv) Last, we have an isomorphism of Hilbert spaces H χ ≃ L ( Q ) induced bythe ˜ G -equivariant isomorphism: C ∞ ( ˜ G ) B → C ∞ ( Q ) , ˆ ϕ ϕ := ˆ ϕ (cid:12)(cid:12) Q , under which, we have Σ = s ⋆e . Proof.
Item (ii) follows from a direct check implying in turn the left- Q -equivariance of s from the left- ˜ G -equivariance of σ . The fact that s q fixes q isolatedlyis a consequence of the following observation. Considering the linear epimorphism .4. UNITARITY AND MIDPOINTS FOR ELEMENTARY SPACES 89 p ⋆ ˜ K : p → q = T e Q ≃ T B ( ˜ G/B ) tangent to the projection p : M ≃ ˜ G/ ˜ K → ˜ G/B ≃ Q , g ˜ K gB , one observes that for every X ∈ p : σ ˜ K⋆B ( p ⋆ ˜ K X ) = ( p ◦ s ˜ K ) ⋆ ˜ K ( X ) = − p ⋆ ˜ K X .
Hence σ ˜ K⋆B = − id q and (i) follows. Last, (iii) and (iv) are immediate consequencesof the Q -equivariant identification Q ≃ ˜ G/B . (cid:3) From now on, we will always make the identification H χ ≃ L ( Q ), under whichwe can derive the action of the individual operators Ω( x ), x ∈ M . For this, we needa preliminary result: Lemma . Let ( M, s, ω ) be a local symplectic symmetric space and let ˆ ϕ ∈ C ∞ ( ˜ G, C ) B be a B -equivariant function. Then, for all q, q ∈ Q and b ∈ B , onehas: L ⋆qb ◦ ˜ σ ⋆ ◦ L ⋆ ( qb ) − ˆ ϕ ( q ) = E ( q − qb ) ˆ ϕ (cid:0) s q q (cid:1) , with (5.11) E ( qb ) := χ (cid:0) C q (cid:0) ˜ σ ( b ) b − (cid:1)(cid:1) , Proof.
A direct computation yields: L ⋆qb ◦ ˜ σ ⋆ ◦ L ⋆ ( qb ) − ˆ ϕ ( q ) = ˆ ϕ ( qb ˜ σ ( b − )˜ σ ( q − q )) = ˆ ϕ ( q ˜ σ ( q − q ) C ˜ σ ( q − q ) ( b ˜ σ ( b − ))) . Under the assumption of locality (Definition 5.30), we have C ˜ σ ( q − q ) ( b ˜ σ ( b − )) ∈ B .The ˜ σ -invariance of χ and item (iii) of Definition 5.30 then yield the formula. (cid:3) Remark . We call the function E in (5.11) the one-point phase . Observethat the latter is well defined thanks to the second condition in the assumption oflocality (Definition 5.30). Corollary . Let ( M, s, ω ) be a local symplectic symmetric space. For ϕ ∈ L ( Q ) and x = qb ˜ K ∈ M , q ∈ Q , b ∈ B , we have Ω( x ) ϕ ( q ) = E ( q − qb ) ϕ (cid:0) s q ( q ) (cid:1) , where E is the phase defined in (5.11) and s is the symmetry of the Lie group Q constructed in Lemma 5.31. In addition to locality (Definition 5.30), we will assume further conditions onthe structure of our polarized symplectic symmetric space (
M, s, ω ), which willenable us to give an explicit expression of the three-point kernel associated to aWKB-quantization of M as well as to prove the triviality of the associated Berezintransform (see Definition 5.46 below). Recall that the notion of midpoint map ona symmetric space is given in Definition 2.10. Definition . A local symplectic symmetric space (
M, s, ω ) is called el-ementary when, within the context of the section 5.2, the following additionalconditions are satisfied:(i) The symmetric space (
Q, s ) is solvable and admits a (necessary unique)midpoint map. (By a result proved in [ ], this implies that Q is expo-nential.) (ii) There exists an exponential Lie subgroup Y of B normalized by Q andsuch that the semi-direct product S := Q ⋉ Y ⊂ ˜ G , acts simply transitively on M .(iii) Denoting by Y the Lie algebra of Y , there exists a global diffeomorphismΨ : Q → q such that h ξ , ( Ad q − − Ad ( s e q ) − ) y i = h ξ , [Ψ( q ) , y ] i , ∀ y ∈ Y , ∀ q ∈ Q . (iv) The Lie algebras Y and q are Lagrangian subspaces of the Lie algebra s of S that are in symplectic duality with respect to the evaluation at theunit element e of S of the symplectic structure transported from M to S via the diffeomorphism S → M = ˜ G/ ˜ K , x x ˜ K . Remark . Let (
M, s, ω ) be an elementary symplectic symmetric space.(i) From now on, we always make the S -equivariant identification: S = Q ⋉ Y → M , qb qb ˜ K .
Observe that under this identification, the ˜ G -invariant Liouville measured M on M is a left Haar measure on S , which under the parametrization g = qb , q ∈ Q , b ∈ Y , is proportional to any product of left invariant Haarmeasures on Q and on Y . We simply denote the latter by d S .(ii) For m a locally essentially bounded Borelian function on Q and x ∈ S , theoperator ˜Ω m ( x ) given in Lemma 5.28 will be simply denoted by Ω m ( x ).This is coherent with the identification above and with the notation ofProposition 5.22 when m = 1. Moreover, the family { Ω m ( x ) } x ∈ S satisfiesthe first axiom, (5.1), of a covariant Moyal quantizer, as given at the verybeginning of this chapter.(iii) Since Q normalizes Y , the restriction to S = Q ⋉ Y of the representation U χ of ˜ G on H χ ≃ L ( Q ) given in (5.9), reads: U χ ( qb ) ϕ ( q ) = χ (cid:0) C q − q ( b ) (cid:1) ϕ ( q − q ) . In the elementary case, we observe the following relation between the one-pointphase E and the diffeomorphism Ψ : Q → q : Lemma . Let ( M, s, ω ) be an elementary symplectic symmetric space. Then,for q ∈ Q and y ∈ Y , we have: E ( q − e y ) = exp { i h ξ, [Ψ( q ) , y ] i} . Proof.
By definition, we have for q ∈ Q and b ∈ Y : E ( q − b ) = χ (cid:0) ˜ σ ( C ˜ σ ( q − ) ( b )) C q − ( b − ) (cid:1) . Next, we write˜ σ ( q − ) = ˜ σ ( q ) − = (cid:0) ˜ σ ( q ) Q ˜ σ ( q ) B (cid:1) − = (cid:0) ˜ σ ( q ) B (cid:1) − (cid:0) s e q (cid:1) − , to get C q − ( b − ) ˜ σ (cid:0) C ˜ σ ( q − ) ( b ) (cid:1) = (cid:0) ˜ σ (cid:0) ˜ σ ( q ) B (cid:1) − (cid:1) ˜ σ (cid:0) C ( s e q ) − ( b ) (cid:1) ˜ σ (cid:0) ˜ σ ( q ) B (cid:1) (cid:0) C q − ( b − ) (cid:1) . .4. UNITARITY AND MIDPOINTS FOR ELEMENTARY SPACES 91 Since Q normalizes Y and B is ˜ σ -stable, we observe that each of the four factors inthe right hand side above, belong to B . Thus, we can split E ( q − b ) according tothis decomposition, to get E ( q − b ) = χ (cid:0) C q − ( b ) (cid:1) χ (cid:0) C ( s e q ) − ( b ) (cid:1) . The result follows from the definition of the diffeomorphism Ψ and the character χ . (cid:3) Now, using Corollary 5.34, Definition 5.27 and Lemma 5.28, under the identi-fication S ≃ M , we note Proposition . Let ( M, s, ω ) be an elementary symplectic symmetric space.Let q ∈ Q , b ∈ Y and let m be an essentially locally bounded Borelian function on Q . Then the densely defined (on B qb = B q –see Definition 5.27 and Lemma 5.28)operator Ω m ( qb ) , acts as: Ω m ( qb ) ϕ ( q ) = m ( q − q ) E ( q − qb ) ϕ ( s q q ) , ∀ ϕ ∈ B q ⊂ H χ , ∀ q ∈ Q .
Corollary . Let ( M, s, ω ) be an elementary symplectic symmetric space, m be an essentially locally bounded Borelian function on Q and f ∈ D ( S ) . Thenthe operator Ω m ( f ) defined by Ω m ( f ) : D ( Q ) → D ′ ( Q ) ,ϕ Ω m ( f ) ϕ := h ψ ∈ D ( Q ) Z Q × S ψ ( q ) f ( qb ) (cid:0) Ω m ( qb ) ϕ (cid:1) ( q ) d S ( qb ) d Q ( q ) i , has a distributional kernel given by Ω m ( f )[ q , q ] = m (cid:0) mid ( e, q − q ) − (cid:1) (cid:12)(cid:12) Jac ( s e ) − (cid:12)(cid:12) ( q − q ) Z Y f ( mid ( q , q ) b ) E (cid:0) mid ( e, q − q ) b (cid:1) d Y ( b ) . Proof.
Observe first that under the decomposition S = Q ⋉ Y , the left Haarmeasure on S coincides with the product of left Haar measures on Q and Y :d S ( qb ) = d Q ( q ) d Y ( b ) , ∀ q ∈ Q , ∀ b ∈ Y . For f ∈ D ( S ) and any Borelian m , it is clear that Ω m ( f ) defines a continuousoperator from D ( Q ) to D ′ ( Q ) and acts as:Ω m ( f ) ϕ ( q ) = Z Q ⋉Y f ( qb ) m ( q − q ) E ( q − qb ) ϕ (cid:0) s q ( q ) (cid:1) d Q ( q ) d Y ( b ) , ϕ ∈ D ( Q ) . For any q ∈ Q , we set q ′ ( q ) := s q ( q ) and we get from the defining property ofthe midpoint map that q = mid ( q , q ′ ). Now observe that left-translations (in thegroup Q ) are automorphisms of the symmetric space ( Q, s ). Indeed, for all q , q, q ′ in Q , we have L q ( s q ( L q − q ′ )) = q q (cid:0) ˜ σ ( q − q − q ′ ) (cid:1) Q = s q q ( q ′ ) . Hence, by Remark 2.12, we get mid ( q , q ′ ) = q mid ( e, q − q ′ ) = L q ◦ ( s e ) − ◦ L q − ( q ′ ) , the invariance of the Haar measure d Q under left translation gives: (cid:12)(cid:12) Jac mid ( q ,. ) (cid:12)(cid:12) ( q ′ ) = (cid:12)(cid:12) Jac ( s e ) − (cid:12)(cid:12) ( q − q ′ ) . Therefore, a direct computation shows that:Ω m ( f ) ϕ ( q ) = Z Q ⋉Y f ( mid ( q , q ) b ) m ( mid ( e, q − q ) − ) (cid:12)(cid:12) Jac ( s e ) − (cid:12)(cid:12) ( q − q ) × E (cid:0) mid ( e, q − q ) b (cid:1) ϕ ( q ) d Q ( q ) d Y ( b ) , and the result follows by identification. (cid:3) Remark . As a consequence of the preceding corollary, we deduce that foran elementary symplectic symmetric space, the second axiom, (5.2), of a covariantMoyal quantizer is satisfied when m ( e ) | Jac ( s e ) − | ( e ) = 1. Indeed, for f ∈ D ( S ), wededuce thatTr (cid:2) Ω m ( f ) (cid:3) = Z Q Ω m ( f )[ q, q ] d Q ( q )= m ( e ) (cid:12)(cid:12) Jac ( s e ) − (cid:12)(cid:12) ( e ) Z Q × Y f ( qb ) E ( b ) d Y ( b ) d Q ( q ) = Z S f ( x ) d S ( x ) , since E ( b ) = 1 for all b ∈ Y by Definition 5.35 (iii) and Lemma 5.37.Our next aim is to understand the geometrical conditions on the functionalparameter m necessary for the quantization map to extend to a unitary operatorfrom L ( M ) to the Hilbert space of Hilbert-Schmidt operators on H χ . For this, weintroduce the following specific function on Q : Definition . For (
M, s, ω ) an elementary symplectic symmetric space,define the function m on Q as:(5.12) m ( q ) := (cid:12)(cid:12) Jac s e ( q − ) Jac Ψ ( q ) (cid:12)(cid:12) / . Remark . Observe that both | Jac Ψ | and | Jac s e | are s ⋆e -invariant. Indeed,we have Ψ ◦ s e = − Ψ and s e ◦ s e = s e ◦ s e , hence the claim follows from | Jac s e | = 1(cf . Lemma 5.31 (iii)). However, m need not be s ⋆e -invariant, as | Jac s e | need not beinvariant under the inversion map on Q . Thus, from Lemma 5.28, the quantizationmap Ω m , need not send real functions to self-adjoint operators.We can now state one of the main results of this chapter: Theorem . Let ( M, s, ω ) be an elementary symplectic symmetric space and m be a Borelian function on Q which is (almost everywhere) dominated by m andassume that Y is Abelian. Then the quantization map: Ω m : f Ω m ( f ) := Z S f ( qb ) Ω m ( qb ) d S ( qb ) , is a bounded operator from L ( S ) to L ( H χ ) with k Ω m k ≤ k m / m k ∞ . Moreover, Ω m is a unitary operator if and only if | m | = m . Proof.
Recall that a linear operator T : D ( Q ) → D ′ ( Q ) extends to a Hilbert-Schmidt operator on L ( Q ), if and only its distributional kernel belongs to L ( Q × Q ). In this case, its Hilbert-Schmidt norm coincides with the L -norm of its kernel. .4. UNITARITY AND MIDPOINTS FOR ELEMENTARY SPACES 93 Thus by Corollary 5.39, we deduce that if f ∈ D ( S ), then the square of the Hilbert-Schmidt norm of Ω m ( f ) reads k Ω m ( f ) k = Z Q × Y | m | ( mid ( e, q − q ) − ) (cid:12)(cid:12) Jac ( s e ) − (cid:12)(cid:12) ( q − q ) f ( mid ( q , q ) b ) f ( mid ( q , q ) b ) × E (cid:0) mid ( e, q − q ) b (cid:1) E (cid:0) mid ( e, q − q ) b (cid:1) d Q ( q ) d Y ( b ) d Q ( q ) d Y ( b ) . Performing the change of variables mid ( q , q ) q (the inverse of the one we per-formed in the proof of Corollary 5.39) and using the relation between the function E and the diffeomorphism Ψ : Q → q given in Lemma 5.37, we get the followingexpression for k Ω m ( f ) k : Z Q × Y | m | ( q − q ) (cid:12)(cid:12) Jac ( s e ) − (cid:12)(cid:12)(cid:0) s e ( q − q ) (cid:1) f ( qb ) f ( qb ) e i h ξ, [Ψ( q − q ) , log( b ) − log( b )] i × d Q ( q ) d Y ( b ) d Q ( q ) d Y ( b ) , and the latter can be rewritten as Z Q × Y | m | ( q ) (cid:12)(cid:12) Jac s e (cid:12)(cid:12)(cid:0) q − (cid:1) f ( qb ) f ( qb ) e i h ξ, [Ψ( q ) , log( b ) − log( b )] i d Q ( q )d Y ( b )d Q ( q ) d Y ( b ) , where in the last line, we used left-invariance of the Haar measure on Q . Setting w = Ψ( q ) ∈ q and d w the Lebesgue measure on q , we get k Ω m ( f ) k = Z Q × q × Y | m | (Ψ − ( w )) f ( qb ) f ( qb ) (cid:12)(cid:12) Jac s e (cid:12)(cid:12)(cid:0) Ψ − ( w ) − (cid:1)(cid:12)(cid:12) Jac Ψ (cid:12)(cid:12)(cid:0) Ψ − ( w ) (cid:1) e i h ξ, [ w , log( b ) − log( b )] i × d Q ( q ) d Y ( b ) d w d Y ( b )= Z Q × q × Y | m | (Ψ − ( w )) m (Ψ − ( w )) f ( qb ) f ( qb ) e − i h ξ, [ w , log( b ) − log( b )] i × d Q ( q ) d Y ( b ) d w d Y ( b )= Z Q × q | m | (Ψ − ( w )) m (Ψ − ( w )) (cid:12)(cid:12)(cid:12)(cid:12)Z Y f ( qb ) e − i h ξ, [ w , log( b )] i d Y ( b ) (cid:12)(cid:12)(cid:12)(cid:12) d Q ( q ) d w . We will next use the relation (which follows from the construction of ξ ∈ ˜ g ⋆ ): h ξ, [ X, Y ] i = ̟ ( X, Y ) , ∀ X, Y ∈ ˜ g , and the fact that q and Y are Lagrangian subspaces in symplectic duality (seeDefinition 5.35). Now, since Y is Abelian and exponential, the exponential map Y → Y : y exp( y ) is a measure preserving diffeomorphism (the vector space Y being endowed with a normalized Lebesgue measure d y ) inducing the isometriclinear identification: exp ⋆ : L ( Y ) → L ( Y ) . Within this set-up, the (isometrical) Fourier transform reads: F : L ( Y ) → L ( q ) , ϕ h w Z Y ϕ ( b ) e − i h ξ, [ w , log( b )] i d Y ( b ) i , where, again, the vector space q is endowed with a normalized Lebesgue measured w . We therefore observe that k Ω m ( f ) k = Z Q × q | m | (Ψ − ( w )) m (Ψ − ( w )) (cid:12)(cid:12) F ( L ⋆q f )( w ) (cid:12)(cid:12) d Q ( q ) d w , where we set L ⋆q f : b f ( qb ). Hence, we see that if | m | ≤ C m , we have: k Ω m ( f ) k ≤ C Z Q × q (cid:12)(cid:12) F ( L ⋆q f )( w ) (cid:12)(cid:12) d w d Q ( q )= C Z Q × Y (cid:12)(cid:12) L ⋆q f ( b ) (cid:12)(cid:12) d Y ( b ) d Q ( q )= C k f k , ∀ f ∈ D ( S ) . i.e . k Ω m ( f ) k ≤ C k f k , ∀ f ∈ D ( S ) . By density, we deduce that Ω m ( f ) is Hilbert-Schmidt for all f ∈ L ( S ) and withequality of norms if and only if | m | = m . In this case, a similar computationshows that for any f , f ∈ L ( S ), we haveTr (cid:2) Ω m ( f ) ∗ Ω m ( f ) (cid:3) = Z S f ( qb ) f ( qb ) d S ( qb ) , which terminates the proof. (cid:3) Remark . Let (
M, s, ω ) be an elementary symplectic symmetric space and m be an essentially bounded function on Q . By Lemma 5.28, we know that when s ⋆e m = m , then the family { Ω m ( x ) } x ∈ S consists of self-adjoint operators. ByRemarks 5.36 (ii) and 5.40 we also know (under a mild normalization condition)that the first two axioms, (5.1) and (5.2), of a covariant Moyal quantizer (as definedat the beginning of this chapter) are satisfied. Now, observe that the content of theprevious Lemma can be summarized as follow:Tr (cid:2) Ω s ⋆e m ( x ) Ω m ( y ) (cid:3) = δ x ( y ) ⇐⇒ | m | = m . Hence, when m is s e -invariant, then the family { Ω m ( x ) } x ∈ S satisfies also the thirdaxiom (5.3). ⋆ -product as the composition law of symbols Definition . Let (
M, s, ω ) be an elementary symplectic symmetric spacesuch that m / m ∈ L ∞ ( Q ) and such that Y is Abelian. Then let σ m : L ( H χ ) → L ( S ) , be the adjoint of the quantization map Ω m . We call the latter the symbol map .Recall that the defining property of the symbol map is h f, σ m [ A ] i = Tr (cid:2) Ω m ( f ) ∗ A (cid:3) , ∀ A ∈ L ( H χ ) , ∀ f ∈ L ( S ) . Hence, the symbol map is formally given by σ m [ A ]( x ) = Tr (cid:2) A Ω s ⋆e m ( x ) (cid:3) , x ∈ S . (5.13)Here again, the trace on the right hand side is understood in the distributionalsense on D ( S ). Note however that when m is essentially bounded, this expressionfor the symbol map genuinely holds on L ( H χ ), the ideal of trace-class operatorson H χ . When m is only locally essentially bounded, it also holds rigorously onthe dense subspace of L ( H χ ), consisting of finite linear combinations of rank oneoperators | ϕ ih ψ | with ψ ∈ D ( Q ) and ϕ ∈ H χ arbitrary. .5. THE ⋆ -PRODUCT AS THE COMPOSITION LAW OF SYMBOLS 95 Definition . Let (
M, s, ω ) be an elementary symplectic symmetric space.Assuming that m / m ∈ L ∞ ( Q ), we then set B m : L ( S ) → L ( S ) , f σ m ◦ Ω m ( f ) , and we call this linear operator the Berezin transform of the quantization mapΩ m . Remark . The Berezin transform measures the obstruction for the symbolmap to be inverse of the quantization map. Said differently, the unitarity of thequantization map on L ( S ) is equivalent to the triviality of the associated Berezintransform. Proposition . Let ( M, s, ω ) be an elementary symplectic symmetric spacewith Y Abelian and let m a Borelian function on Q such that m / m ∈ L ∞ ( Q ) andsuch that Y is Abelian. Then (i) The Berezin transform is a positive and bounded operator on L ( S ) , with k B m k ≤ k m / m k ∞ . (ii) The Berezin transform is a kernel operator with distributional kernel givenby B m [ x , x ] = Tr (cid:2) Ω s ⋆e m ( x ) Ω m ( x ) (cid:3) , and the latter can be identified with B m [ x , x ] = δ q ( q ) × Z q | m | m (cid:0) Ψ − ( w ) (cid:1) e i h ξ, [ w, log b − log b ] i d w , where x j = q j b j ∈ S , j = 1 , . Proof.
Note that by construction B m = Ω ∗ m ◦ Ω m , yielding positivity andboundedness from boundedness of Ω m . The operator norm estimate comes fromthose of Theorem 5.43. The second claim comes from the computation done in theproof of this Theorem. (cid:3) For m = m , the symbol map σ m is the inverse of the quantization map Ω m . Inparticular, the associated Berezin transform is trivial. A ˜ G -equivariant associativeproduct ⋆ m on L ( S ) is then defined: f ⋆ m f := σ m (cid:2) Ω m ( f ) Ω m ( f ) (cid:3) , ∀ f , f ∈ L ( S ) . (5.14)We deduce from (5.13) that the product (5.14) is a three-point kernel product, withdistributional kernel given by the operator trace of a product of three Ω’s: f ⋆ m f ( x ) = Z S × S f ( y ) f ( z ) Tr (cid:2) Ω s ⋆e m ( x ) Ω m ( y ) Ω m ( z ) (cid:3) d S ( y ) d S ( z ) . We will return to the explicit form of the three-point kernel and its geometricinterpretation in the next section.We now come to an important point. Putting together Remark 5.42 and The-orem 5.43, we see that in general the quantization map Ω m need not be unitaryand involution preserving (the complex conjugation on L ( S ) and the adjoint on L ( H χ )) at the same time. However, in most cases (e.g . for elementary normal j -groups), the function m is s e -invariant, which implies that the complex conju-gation is an involution of the Hilbert algebra (cid:0) L ( S ) , ⋆ m (cid:1) : Proposition . Let ( M, s, ω ) be an elementary symplectic symmetric spacewith Y Abelian. Assuming further that s ⋆e m = m , then for all f , f ∈ L ( S ) , wehave: f ⋆ m f = f ⋆ m f . Next, we pass to a possible approach to define a ⋆ -product for the quantizationmap Ω m in the more general context of an arbitrary function m (and without theassumption that Y is Abelian). Definition . Let (
M, s, ω ) be a polarized, local and elementary symplecticsymmetric space and fix m a locally essentially bounded Borelian function on Q .We then let L m ( S ), be the Hilbert-space of classes of measurable functions on S forwhich the norm underlying the following scalar product is finite : h f , f i m := Z Q × q | m | m (cid:0) Ψ − ( w ) (cid:1) (cid:16) Z Y f ( qb ) e − i h ξ, [ w, log b ] i d Y ( b ) (cid:17) × (cid:16) Z Y f ( qb ) e i h ξ, [ w, log b ] i d Y ( b ) (cid:17) d Q ( q ) d w . Remark . Formally, we have h f , f i m = h f , B m f i = Tr (cid:2) Ω m ( f ) ∗ Ω m ( f ) (cid:3) , where h ., . i denotes the inner product of L ( S ) and Tr is the operator trace on H χ .Repeating the computations done in the proof of Theorem 5.43, we deducefollowing extension of the latter: Proposition . Let ( M, s, ω ) be an elementary symplectic symmetric spaceand let m be a locally essentially bounded Borelian function on Q . (i) The quantization map Ω m is a unitary operator from L m ( S , d S ) to L ( H χ ) , (ii) Associated to the quantization map Ω m , there is a deformed product ⋆ m on L m ( S , d S ) , which is formally given by: f ⋆ m f = B − m ◦ σ m (cid:2) Ω m ( f )Ω m ( f ) (cid:3) , (iv) (cid:0) L m ( S ) , ⋆ m (cid:1) is a Hilbert algebra and the complex conjugation is an invo-lution when s ⋆e m = m . The aim of this section is to compute the distributional three-point kernelTr (cid:2) Ω m ( x ) Ω m ( y ) Ω m ( z ) (cid:3) of the product ⋆ m given in (5.14). We start with twopreliminary results extracted from [ ]: Theorem . Let ( M, s, ω ) be an elementary symplectic symmetric space.Given three points q , q , q in Q , the equation s q s q s q ( q ) = q , admits a unique solution q ≡ q ( q , q , q ) ∈ Q . In particular, this yields a well-defined map Q → Q : ( q , q , q ) ( q, s q ( q ) , s q s q ( q )) . This is however not the approach we will follow for the symmetric spaces underlying ele-mentary normal j -groups–see Proposition 6.19. We do not exclude the possibility that L m ( S ) be trivial. .6. THE THREE-POINT KERNEL 97 The latter is a global diffeomorphism called the medial triangle map whose in-verse is given by: Φ Q : Q → Q , ( q , q , q ) (cid:0) mid ( q , q ) , mid ( q , q ) , mid ( q , q ) (cid:1) .x x x mid ( x , x ) mid ( x , x ) mid ( x , x ) •• •• •• Figure 1.
The medial triangle of three points
Lemma . Let us consider the map ν : Q → Q : ( q , q , q ) ( mid ( e, q − q ) , mid ( e, q − q ) , mid ( e, q − q )) . Then ν ◦ Φ − Q ( q , q , q ) = ( q − q , ( s q q ) − q , ( s q s q q ) − q ) , where q is the unique fixed point of s q s q s q . Proof.
We have mid ( e, q − q ) = q − mid ( q , q ) and similarly for the otherpoints. Therefore setting( q ′ , q ′ , q ′ ) := (cid:0) mid ( q , q ) , mid ( q , q ) , mid ( q , q ) (cid:1) = Φ Q ( q , q , q ) , we get ν ( q , q , q ) = ( q − q ′ , q − q ′ , q − q ′ ) . A quick look at the above figure leads to s q ′ s q ′ s q ′ ( q ) = q and the assertionimmediately follows. (cid:3) Theorem . Let ( M, s, ω ) be an elementary symplectic symmetric spacewith Y Abelian. Assume that
Jac s e is invariant under the inversion map on Q . Let q ≡ q ( q , q , q ) denote the unique solution of the equation s q s q s q ( q ) = q in Q .Set J ( q , q , q ) := | Jac ( s e ) − | ( q − q ) | Jac ( s e ) − | ( q − q ) | Jac ( s e ) − | ( q − q ) . Then, one has: s ⋆e m = m , and the kernel of the product ⋆ m (5.14) is given by: K m ( x , x , x ) = J / (Φ − Q ( q , q , q )) (cid:12)(cid:12) Jac Φ − Q (cid:12)(cid:12) ( q , q , q ) × | Jac Ψ | / (cid:0) q − q (cid:1) | Jac Ψ | / (cid:0) q − s q q (cid:1) | Jac Ψ | / (cid:0) q − s q s q q (cid:1) × E (cid:0) q − x (cid:1) E (cid:0) ( s q q ) − x (cid:1) E (cid:0) ( s q s q q ) − x (cid:1) . Proof.
Under the assumption that Jac s e is invariant under the inversion mapon Q , m is s e -invariant and reads: m ( q ) = (cid:12)(cid:12) Jac s e ( q ) Jac Ψ ( q ) (cid:12)(cid:12) / . Accordingly, for f ∈ L ( S ), we get from Corollary 5.39 the following expression forthe operator kernel of Ω m ( f ):Ω m ( f )[ q , q ] = | Jac Ψ | / (cid:0) mid ( e, q − q ) − (cid:1) (cid:12)(cid:12) Jac ( s e ) − (cid:12)(cid:12) / ( q − q ) × Z Y f ( mid ( q , q ) b ) E (cid:0) mid ( e, q − q ) b (cid:1) d Y ( b ) . Since for f ∈ L ( S ), Ω m ( f ) is Hilbert-Schmidt, the product of three Ω m ( f )’s is afortiori trace-class and thus we can employ the formula:Tr h Ω m ( f )Ω m ( f )Ω m ( f ) i = Z Q Ω m ( f )[ q , q ] Ω m ( f )[ q , q ] Ω m ( f )[ q , q ] d Q ( q ) d Q ( q ) d Q ( q ) . Using Theorem 5.53 and the formula above for the kernel of Ω m ( f j ), we see thatthe above trace equals: Z Q × Y | Jac Ψ | / (cid:0) mid ( e, q − q ) − (cid:1) | Jac Ψ | / (cid:0) mid ( e, q − q ) − (cid:1) × | Jac Ψ | / (cid:0) mid ( e, q − q ) − (cid:1) J / ( q , q , q ) f ( mid ( q , q ) b ) f ( mid ( q , q ) b ) × f ( mid ( q , q ) b ) E (cid:0) mid ( e, q − q ) b (cid:1) E (cid:0) mid ( e, q − q ) b (cid:1) E (cid:0) mid ( e, q − q ) b (cid:1) × d Q ( q ) d Q ( q ) d Q ( q ) d Y ( b ) d Y ( b ) d Y ( b ) . Performing the change of variable ( q , q , q ) Φ − Q ( q , q , q ) and setting x j := q j b j ∈ S , j = 0 , ,
2, we get Z S | Jac Ψ | / (cid:0) q − q (cid:1) | Jac Ψ | / (cid:0) q − s q q (cid:1) | Jac Ψ | / (cid:0) q − s q s q q (cid:1) J / (Φ − Q ( q , q , q )) × (cid:12)(cid:12) Jac Φ − Q (cid:12)(cid:12) ( q , q , q ) E (cid:0) q − x (cid:1) E (cid:0) ( s q q ) − x (cid:1) E (cid:0) ( s q s q q ) − x (cid:1) f ( x ) f ( x ) × f ( x ) d S ( x ) d S ( x ) d S ( x ) , where q ≡ q ( q , q , q ) is the unique solution of the equation s q s q s q ( q ) = q (seeLemma 5.54). The result then follows by identification. (cid:3) Last, using Lemma 5.37, we deduce the following expression for the phase inthe kernel of the product ⋆ m : Corollary . Write K m = A m e − iS for the three-point kernel given inProposition 5.55. Then we have for x j = q j b j ∈ S , q j ∈ Q , b j ∈ Y , j = 0 , , : S ( x , x , x ) = h ξ, [Ψ( q − q ) , log b ] i + h ξ, [Ψ( q − s q q ) , log b ] i + h ξ, [Ψ( q − s q s q q ) , log b ] i , where q ≡ q ( q , q , q ) is the unique solution of the equation s q s q s q ( q ) = q (seeTheorem 5.53). .7. EXTENSIONS OF POLARIZATION QUADRUPLES 99 We first observe that, given two polarization quadruples ( G j , σ j , ξ j , B j ), j = 1 , φ between them yields a G -equivariant intertwiner:(5.15) φ ⋆ : C ∞ ( G ) B → C ∞ ( G ) B , such that U χ ( g ) φ ⋆ ˆ ϕ = φ ⋆ (cid:0) U χ ( φ ( g )) ˆ ϕ (cid:1) . Note that the condition φ ⋆ ξ = ξ (cf . Definition 5.15) implies χ ( b ) = χ ( φ ( b ))in view of (5.7).In the context of the transvection and full polarization quadruples, we observe: Lemma . Let ( M, s, ω ) be an elementary symplectic symmetric space, asso-ciated to a non-exact polarized transvection triple ( g , σ, ̟ ) . Consider ( L , σ L , ξ, B ) and ( ˜ G, ˜ σ, ξ (cid:12)(cid:12) ˜ G , B ) and the full and transvection polarization quadruples as in Defi-nition 5.19. Then, the intertwiner (5.15) corresponding to the injection ˜ G → L isa linear isomorphism. Proof.
The injection j : ˜ G → L induces a global diffeomorphism ˜ G/B → L / B , gB g B . Indeed, the map ˜ G/ ˜ K → L / ˜ D : g ˜ K g ˜ D is an identification.Considering the natural projections ˜ G/ ˜ K → ˜ G/B and L / ˜ D → L / B , one observesthat the diagram ˜ G/ ˜ K −→ L / ˜ D ↓ ↓ ˜ G/B φ −→ L / B , where φ ( gB ) := g B , is commutative. In particular, φ is surjective. Examining itsdifferential proves that is also a submersion. The space Q = ˜ G/B , being expo-nential, has trivial fundamental group. The map φ is therefore a diffeomorphism.Also the restrictions C ∞ ( ˜ G ) B → C ∞ ( Q ) and C ∞ ( L ) B → C ∞ ( Q ) are linear iso-morphisms and one observes that j ⋆ ˆ ϕ | Q = ˆ ϕ | Q . (cid:3) Note that when the modular function of B coincides with the restriction to B ofthe modular function of L , then there exists a L -invariant measure on L / B . Fromthe isomorphism L / B ≃ ˜ G/B ≃ Q , we see that the later is a left-Haar measure on Q . Hence, under the assumption above, we deduce that the left-Haar measure d Q is also L -invariant. This, together with the above Lemma, yields: Lemma . In the setting of Lemma 5.57 and when the modular function of B coincides with the restriction to B of the modular function of L , the injection ˜ D → L induces a unitary representation R : ˜ D → U ( H χ ) of the corresponding analyticsubgroup ˜ D ⊂ L on the representation space H χ associated to the transvectionquadruple ( ˜ G, ˜ σ, ξ, B ) . Consider now two Lie groups G j , j = 1 ,
2, with unitary representations ( U j , H j ),together with a Lie group homomorphism ρ : G → G , and form the associated semi-direct product G ⋉ R G , where(5.16) R g ( g ) := C ρ ( g ) ( g ) = ρ ( g ) g ρ ( g ) − , ∀ g j ∈ G j .
00 5. QUANTIZATION OF POLARIZED SYMPLECTIC SYMMETRIC SPACES
We deduce the representation homomorphism: R : G → U ( H ) , g U ( ρ ( g )) . (5.17)Within this setting, we first observe: Lemma . Parametrizing an element g ∈ G ⋉ R G as g = g .g , the map U : G ⋉ R G → U ( H ⊗H ) , g U ( g ) ⊗ R ( g ) U ( g ) , defines a unitary representation of G ⋉ R G on the tensor product Hilbert space H := H ⊗H . Proof.
Let g j , g ′ j ∈ G j , j = 1 ,
2. Then, on the first hand: U ( g g .g ′ g ′ ) = U (cid:0) g g ′ R g ′ − ( g ) g ′ (cid:1) = U ( g g ′ ) ⊗ R ( g g ′ ) U (cid:0) R g ′ − ( g ) g ′ (cid:1) = U ( g g ′ ) ⊗ R ( g ) U ( g ) R ( g ′ ) U ( g ′ ) , while on the second hand: U ( g g ) U ( g ′ g ′ ) = (cid:0) U ( g ) ⊗ R ( g ) U ( g ) (cid:1) (cid:0) U ( g ′ ) ⊗ R ( g ′ ) U ( g ′ ) (cid:1) = U ( g g ′ ) ⊗ R ( g ) U ( g ) R ( g ′ ) U ( g ′ ) , and the proof is complete. (cid:3) Consider Lastly two full polarization quadruples ( L j , σ j , ξ j , B j ), j = 1 ,
2, asso-ciated to two local symplectic symmetric spaces ( M j , s j , ω j ). Let also ( U χ j , H χ j , Ω j )be the unitary representation of L j and of the representation of the symplecticsymmetric space M j = ˜ G j / ˜ K j = L j / ˜ D j (see Remark 5.29). Finally, let K bea Lie subgroup of L that acts transitively on M , and consider the associated K -equivariant diffeomorphism: ϕ : K / ( K ∩ ˜ D ) → M . Now, given a Lie group homomorphism ρ : K → ˜ D ⊂ L , we can form the semi-direct product K ⋉ R L , according to (5.16). Under these conditions, we have theglobal identification: (cid:0) K ⋉ R L (cid:1) / (cid:0) ( K ∩ ˜ D ) ⋉ R ˜ D (cid:1) → M × M , (5.18) ( g .g )( K ∩ ˜ D ) ⋉ R ˜ D (cid:0) ϕ ( g K ∩ ˜ D ) , g ˜ D (cid:1) . Proposition . Let U be the unitary representation of K ⋉ R L on H := H χ ⊗ H χ constructed in Lemma 5.59. Then under the conditions displayed above, (i) the map Ω : K ⋉ R L → U sa ( H ) , g U ( g ) ◦ (Σ ⊗ Σ ) ◦ U ( g ) ∗ , is constant on the left cosets of ( K ∩ ˜ D ) ⋉ R ˜ D in K ⋉ R L . (ii) For every g ∈ K and g ∈ L , one has Ω ( g .g ) = Ω ( g ) ⊗ Ω ( g ) . (iii) Under the identification (5.18) , the quotient map Ω : M × M → U sa ( H ) , is K ⋊ R L -equivariant. Warning: the reverse order in the group elements. .7. EXTENSIONS OF POLARIZATION QUADRUPLES 101
Proof.
We start by checking item (ii). Observe first that U ( g g ) = U ( g R g − g ) = U χ ( g ) ⊗ R ( g ) U χ ( R g − g )= U χ ( g ) ⊗ U χ ( g ) R ( g ) . Now, since ρ is ˜ D -valued, we have for all g ∈ K : R ( g )Σ R ( g ) ∗ = U χ ( ρ ( g ))Σ U χ ( ρ ( g )) ∗ = Σ , (cf . Lemma 5.11). Moreover, one has Ω ( g g ) = ( U χ ( g ) ⊗ U χ ( g ) R ( g )) ◦ (Σ ⊗ Σ ) ◦ ( U χ ( g ) ∗ ⊗ R ( g ) ∗ U χ ( g ) ∗ )= U χ ( g )Σ U χ ( g ) ∗ ⊗ U χ ( g ) R ( g )Σ R ( g ) ∗ U χ ( g ) ∗ = Ω ( g ) ⊗ Ω ( g ) . This implies (ii) and (i) consequently. Regarding item (iii), one observes at thelevel of K ⋉ R L that Ω ( gg ′ ) = U ( g ) Ω ( g ′ ) U ( g ) ∗ , which is enough to conclude. (cid:3) Remark . In the same manner as in Definition 5.27, given a Borelianfunction m on the product manifold ( L / ˜ D ) × ( L / ˜ D ), we may define for g ∈ K ⋉ R L : Ω m ( g ) := U ( g ) ◦ m ◦ (Σ ⊗ Σ ) ◦ U ( g ) ∗ . Of course, the above procedure can be iterated, namely one observes:
Proposition . Let ( L j , σ j , ξ j , B j ) , j = 1 , . . . , N , be N full polarizationquadruples, associated to N elementary symplectic symmetric spaces ( M j , s j , ω j ) satisfying the extra conditions of coincidence of the modular function on B j withthe restriction to B j of the modular function of L j , according to Lemmas 5.57 and5.58. For every j = 1 , . . . , N − , consider a subgroup K j that acts transitively on M j together with a Lie group homomorphism ρ j : K j → ˜ D j +1 . Set K N := L N ,assume that for every such j , the subgroup ρ j ( K j ) normalizes K j +1 in L j +1 anddenote by R j the corresponding homomorphism from K j to Aut ( K j +1 ) . Then,iterating the procedure described in Proposition 5.60 yields a map Ω : M × M × · · · × M N → U sa ( H ) , into the self-adjoint unitaries on the product Hilbert space H := H χ ⊗ · · · ⊗ H χ N that is equivariant under the natural action of the Lie group ( . . . (( K ⋉ R K ) ⋉ R K ) ⋉ R . . . ) ⋉ R N − L N . This ‘elementary’ tensor product construction for the quantization map on di-rect products of polarized symplectic symmetric spaces (but with covariance undersemi-direct products of subgroups of the covariance group of each piece) allows totransfer most of the results of the previous sections. For notational convenience,we formulate all that follows in the context of two elementary pieces, i.e . in thecontext of Proposition 5.60 rather than in the context of Proposition 5.62.So in all that follows, we assume we are given two elementary symplectic sym-metric spaces ( M j , s j , ω j ), j = 1 , S j = Q j ⋉ Y j , j = 1 ,
2, be the subgroups of ˜ G j (and thus of L j ) that acts simply transitivelyon M j . We also assume that we are given a homomorphism ρ : S → ˜ D (i.e .
02 5. QUANTIZATION OF POLARIZED SYMPLECTIC SYMMETRIC SPACES the role of K in Proposition 5.60 is played by S ). In this particular context, theidentification (5.18) becomes: S ⋉ R S → M × M , g .g ( g ˜ D , g ˜ D ) . We also let m := m ⊗ m be the smooth function on Q × Q , where m j isthe function on Q j given in (5.12). Combining Proposition 5.60 with the results ofsections 5.3, 5.4, 5.5 and 5.6, we eventually obtain: Theorem . Let ( M j , s j , ω j ) , j = 1 , , be two elementary symplectic sym-metric spaces and consider an homomorphism ρ : S → ˜ D . Within the notationsgiven above, we have: (i) Identifying H χ ⊗H χ with L ( Q × Q ) and parametrizing an element g ∈ S ⋉ R S as g = q b q b , q j ∈ Q j , b j ∈ Y j , we have for ϕ ∈ D ( Q × Q ) : Ω m ( g ) ϕ (¯ q , ¯ q ) = m ( q − ¯ q ) m ( q − ¯ q ) E S ⋉S (cid:0) ¯ q − q b , ¯ q − q b ) ϕ ( s q ¯ q , s q ¯ q ) , where E S ⋉S (cid:0) q b , q b ) := E S (cid:0) q b ) E S ( q b ) , ∀ q j ∈ Q j , ∀ b j ∈ Y j , and E S j , j = 1 , , is the one-point phase attached to each elementarysymplectic symmetric space M j as given in Lemma 5.32. (ii) Moreover, when Y and Y are Abelian, the map Ω m : L ( S ⋉ R S ) → L (cid:0) H χ ⊗ H χ (cid:1) ,f Z S ⋉ R S f ( g ) Ω m ( g ) d S ⋉ R S ( g ) , is unitary and S ⋉ R S equivariant. (iii) Denoting by σ m the adjoint of Ω m , the associated deformed product: f ⋆ m f := σ m (cid:2) Ω m ( f ) Ω m ( f ) (cid:3) , takes on D ( S ⋉ R S ) the expression Z ( S ⋉ R S ) K S ⋉ R S m ( g, g ′ , g ′′ ) f ( g ′ ) f ( g ′′ ) d S ⋉ R S ( g ′ ) d S ⋉ R S ( g ′′ ) , where the three-points kernel K S ⋉ R S m is given, with g = g g , g ′ = g ′ g ′ and g ′′ = g ′′ g ′′ by K S ⋉ R S m ( g, g ′ , g ′′ ) := K S m ( g , g ′ , g ′′ ) K S m ( g , g ′ , g ′′ ) , with K S j m , j = 1 , , as given in Proposition 5.55. HAPTER 6
Quantization of K¨ahlerian Lie groups
The aim of this chapter is two-fold. First, we establish that the symplecticsymmetric space ( S , s, ω S ) associated with an elementary normal j -group S (seesection 2.2) underlies an elementary local and polarized symplectic symmetric space,in the sense of Definitions 5.19, 5.30 and 5.35. Using the whole construction ofchapter 5, we will then be able to construct a B -equivariant quantization map forany normal j -group B . Second, we will show that the composition law of symbolsassociated to this quantization map, coincides exactly with the left- B -equivariantstar-products on B that we have considered in chapter 3. We start by describing the non-exact transvection siLa ( g , σ, ̟ ) underlying thesymplectic symmetric space structure ( S , s, ω S ) of an elementary normal j -group,as described in section 2.2.The transvection group G of S is the connected and simply connected Liegroup whose Lie algebra g is a one-dimensional split extension of two copies of theHeisenberg algebra:(6.1) g := a ⋉ ρ ( h ⊕ h ) , where, again, a = R H and the extension homomorphism is given by ρ := ρ h ⊕ ( − ρ h ) ∈ Der( h ⊕ h ), with ρ h defined in (2.1). The involution σ of g is given by(6.2) σ (cid:0) aH + ( X ⊕ Y ) (cid:1) := ( − aH ) + ( Y ⊕ X ) , ∀ a ∈ R , ∀ X, Y ∈ h . One has the associated ( ± g = k ⊕ p , k := h + and p := a ⊕ h − , where for every subspace F ⊂ h , we set(6.3) F ± := { X ⊕ ( ± X ) , X ∈ F } ⊂ h ⊕ h , and for every element X ∈ h we let X ± := ( X ⊕ ( ± X )) ∈ h ⊕ h . Last, we define ̟ ∈ Λ g ⋆ by(6.4) ̟ ( H, E − ) = 2 and ̟ ( v − , v ′− ) = ω ( v, v ′ ) , ∀ v, v ′ ∈ V , and by zero everywhere else on g × g . Note that ̟ is k -invariant and its restrictionto p is non-degenerate. This implies that ̟ is a Chevalley two-cocycle (see [ ]).Also, from [ H, h − ] = h + = k , we deduce that [ p , p ] = k and clearly the action of k on p is faithful. Thus, in terms of Definition 5.4, we have proved the following: Proposition . The siLa ( g , σ, ̟ ) defined by (6.1) , (6.2) and (6.4) , is atransvection symplectic triple. Consider now (˜ g , ˜ σ, δξ ) the exact siLa constructed out of the non-exact siLa( g , σ, ̟ ) as in Lemma 5.11. Recall that ˜ g is the one-dimensional central extensionof g with generator Z and table[ X, Y ] ˜ g = [ X, Y ] g + ̟ ( X, Y ) Z , ∀ X, Y ∈ g . The involution ˜ σ equals id ˜ k ⊕ ( − id p ), where ˜ k = k ⊕ R Z and ξ ∈ ˜ g ⋆ is defined by h ξ, Z i = 1 and ξ (cid:12)(cid:12) g = 0. Accordingly, we set ˜ G = exp { ˜ g } and ˜ K = exp { ˜ k } . Weidentify ˜ g with ˜ G via the global chart:(6.5) aH + v ⊕ v + t E ⊕ t E + ℓZ exp { aH } exp { v ⊕ v + t E ⊕ t E + ℓZ } , where a, t , t , ℓ ∈ R and v , v ∈ V . The group law of ˜ G in these coordinates thenreads: ( a, v , v , t , t , ℓ )( a ′ , v ′ , v ′ , t ′ , t ′ , ℓ ′ ) = (cid:16) a + a ′ , e − a ′ v + v ′ , e a ′ v + v ′ ,e − a ′ t + t ′ + e − a ′ ω ( v , v ′ ) , e a ′ t + t ′ + e a ′ ω ( v , v ′ ) ,ℓ + ℓ ′ + ( e − a ′ − t + ( e a ′ − t + ω ( e − a ′ v − e a ′ v , v ′ − v ′ ) (cid:17) , (6.6)and the inversion map is given by:( a, v , v , t , t , ℓ ) − = (cid:0) − a, − e a v , − e − a v , − e a t , − e − a t , − ℓ − ( e a − t − ( e − a − t (cid:1) . Moreover the involution ˜ σ admits the following expression:˜ σ ( a, v , v , t , t , ℓ ) = ( − a, v , v , t , t , ℓ ) . Under the parametrization of ˜ G given above, we consider the following global co-ordinates system on ˜ G/ ˜ K :(6.7) ˜ G/ ˜ K → R d +2 , ( a, v , v , t , t , ℓ ) e K (cid:0) a, v − v , t − t − ω ( v , v ) (cid:1) . From the formula s g ˜ K ( g ′ ˜ K ) = g ˜ σ ( g − g ′ ) ˜ K for the symmetry on ˜ G/ ˜ K , we deducethe following isomorphism of symplectic symmetric spaces: Proposition . Under the identifications S ≃ R d +2 ≃ ˜ G/ ˜ K associated withthe charts (2.4) and (6.7) , the symplectic symmetric space ˜ G/ ˜ K underlying theexact siLa (˜ g , ˜ σ, δξ ) defined above, is isomorphic to the symplectic symmetric space ( S , s, ω S ) underlying an elementary normal j -group S , as given in section 2.1. Next, we need to endow ( S , s, ω S ) with a structure of polarized symplectic sym-metric space. From Lemma 5.9, it suffices to specify W , a k -invariant Lagrangiansubspace of p . To this aim, we again consider the splitting of the 2 d -dimensionalsymplectic vector space ( V, ω ) into a direct sum of two Lagrangian subspaces insymplectic duality: V = l ⋆ ⊕ l . Relatively to this decomposition and within the notation (6.3), we define: W := l − ⊕ R E − ⊂ g . Following then Proposition 5.18, we let b := ˜ k ⊕ W = k ⊕ R Z ⊕ W , .1. THE TRANSVECTION QUADRUPLE OF AN ELEMENTARY NORMAL j -GROUP 105 be the polarization Lie algebra. Accordingly with the terminology introduced inDefinition 5.19, we call (˜ g , ˜ σ, ξ, b ) the transvection quadruple of the symplecticsymmetric space ( S , s, ω S ).Regarding the question of existence of ˜ G -invariant measures on the homoge-neous spaces ˜ G/ ˜ K and ˜ G/B , we first observe the following fact:
Lemma . Let ( g = k ⊕ p , σ ) be the involutive Lie algebra associated to asolvable, simply connected, oriented symmetric space M = G/K such that [ p , p ] = k .Then both G and K are unimodular Lie groups. Proof.
Under the orientation hypothesis, let ν p denote a K -invariant volumeelement on p ≃ T K M . Since k = [ p , p ] ⊂ [ g , g ], under the solvability assumption,the Lie algebra k is nilpotent. Therefore for every Z ∈ k , one has Tr ( ad Z | k ) = 0 andthere exists an ad -invariant volume element ν k on k . The volume element ν k ∧ ν p on g is therefore k -invariant. It is also ad p -invariant since, due to the iLa condition, forevery X ∈ p , the element ad X is trace-free. The simple-connectedness of M impliesthe connectedness of K . The latter is therefore unimodular, as well as G . (cid:3) Remark . The above lemma implies that ˜ G , ˜ K and B := exp { b } are allunimodular ( b is nilpotent). In particular, there exist ˜ G -invariant measures on thehomogeneous spaces ˜ G/ ˜ K and ˜ G/B .Last, we need to specify the local and elementary structures underlying thepolarized symplectic symmetric space ( S , s, ω S ), as introduced in Definition 5.30and Definition 5.35 respectively. We first note: Lemma . Let q := a ⊕ ( l ⋆ ⊕ and Y := ( l ⊕ ⊕ R (cid:0) ( E ⊕
0) + Z (cid:1) . Then q is a Lie subalgebra of ˜ g supplementary to b and Y is an Abelian Lie subal-gebra of b which is normalized by q . Moreover, the associated semi-direct product q ⋉ Y is naturally isomorphic to the Lie algebra s and induces the vector spacedecomposition ˜ g = s ⊕ ˜ k . Proof.
First observe that for all X ∈ h , one has X ⊕ X − + X + ∈ h ⊕ h and therefore ̟ ( H, E ⊕
0) = ̟ ( H, E − + E + ) = ̟ ( H, E − ) = 2 ,̟ ( v ⊕ , v ′ ⊕
0) = ̟ ( v − + v + , v ′− + v ′ + ) = ̟ ( v − , v ′− ) = ω ( v, v ′ ) , ∀ v, v ′ ∈ V .
The fact that q is a Lie subalgebra follows from[ H, l ⋆ ⊕ ˜ g = [ H, l ⋆ ⊕
0] + ̟ ( H, l ⋆ ⊕ Z = l ⋆ ⊕ , and [ l ⋆ ⊕ , l ⋆ ⊕ ˜ g = ω ( l ⋆ , l ⋆ ) (cid:0) E ⊕ Z (cid:1) = 0 . Next, observe that Y is Abelian:[ Y , Y ] ˜ g = [ l ⊕ , l ⊕ ˜ g + [ l ⊕ , R (cid:0) ( E ⊕
0) + Z (cid:1) ] ˜ g = ω ( l , l ) (cid:0) ( E ⊕
0) + Z (cid:1) = 0 . To see that q normalizes Y , let x ∈ l and t ∈ R . Then one has[ H, x ⊕ t ( E ⊕ Z )] ˜ g = [ H, x ⊕
0] + ̟ ( H, x − ) Z + t [ H, E ⊕
0] + t̟ ( H, E − ) Z = x ⊕ t ( E ⊕ Z ) .
06 6. QUANTIZATION OF K¨AHLERIAN LIE GROUPS
Similarly, for all y ∈ l ⋆ , one has:[ y ⊕ , x ⊕ t ( E ⊕ Z )] ˜ g = [ y ⊕ , x ⊕
0] + ω ( y, x ) Z = ω ( y, x )( E ⊕ Z ) . The rest of the statement is immediate. (cid:3)
Remark . Neither q nor Y are ˜ σ -stable. However, since ˜ σ ( q ) = a ⊕ (0 ⊕ l ⋆ )and [0 ⊕ l ⋆ , Y ] = 0, one sees that ˜ σ ( q ) normalizes Y as well. Lemma . Equipped with the subgroup Q = exp { q } of ˜ G , the symplecticsymmetric space ( S , s, ω S ) is local in the sense of Definition 5.30. Proof.
Note first that b = k ⊕ W ⊕ R Z = h + ⊕ l − ⊕ R E − ⊕ R Z = l ⋆ + ⊕ ( l ⊕ l ) ⊕ ( R E ⊕ R E ) ⊕ R Z .
Thus, under the parametrization (6.5) of ˜ G , we have(6.8) B = (cid:8) (0 , n ⊕ m , n ⊕ m , t , t , ℓ ) : m , m ∈ l , n ∈ l ⋆ , t , t , ℓ ∈ R (cid:9) , and Q = (cid:8) ( a, n, , , ,
0) : n ∈ l ⋆ , a ∈ R (cid:9) . Thus for q = ( a, n, , , ,
0) and b = (0 , n ′ ⊕ m ′ , n ′ ⊕ m ′ , t ′ , t ′ , ℓ ′ ) ∈ B , we haveusing (6.6): q.b = (cid:0) a, ( n + n ′ ) ⊕ m ′ , n ′ ⊕ m ′ , t ′ + ω ( n, m ′ ) , t ′ , ℓ ′ + ω ( n, m ′ − m ′ ) (cid:1) , from which we deduce that the map Q × B → ˜ G , ( q, b ) q.b , is a global diffeomorphism (i.e . the first condition of Definition 5.30 is satisfied).Note that identifying B with b , one has b ˜ σ ( b − ) = 2 b p , where we set b =: b ˜ k + b p according to the vector space decomposition b = ˜ k ⊕ ( b ∩ p ).For the second condition, observe that as b ∩ p = l − ⊕ R E − , we get[ a , b ∩ p ] = [ a , l − ] ⊕ [ a , R E − ] = l + ⊕ R E + ⊂ h + = k ⊂ b . To check the last condition, consider q = ( a, n, , , , ∈ Q , with a ∈ R , n ∈ l ⋆ .We then have˜ σq = ( − a, , n, , ,
0) = ( − a, − n, , , , , n, n, , ,
0) = (˜ σq ) Q (˜ σq ) B , (6.9)since (0 , n, n, , , ∈ l ⋆ + ⊂ h + ⊂ b . Thus, χ (cid:0) (˜ σq ) B (cid:1) = 1 since for b = (0 , n ⊕ m , n ⊕ m , t , t , ℓ ) ∈ B , we have χ ( g ) = e iℓ . (cid:3) Next, we come to the symmetric space structure of the group Q : Lemma . In the global chart: (6.10) q ≃ a ⊕ l ⋆ → Q , ( a, n ) exp { aH } exp { n ⊕ } , the left invariant symmetric space structure s on Q described in Lemma 5.31 reads: (6.11) s ( a,n ) ( a ′ , n ′ ) = (cid:0) a − a ′ , a − a ′ ) n − n ′ (cid:1) . Moreover, the symmetric space ( Q, s ) admits a midpoint map, which in the coordi-nates above, is given by: mid (cid:0) ( a, n ) , ( a , n ) (cid:1) = (cid:16) a + a , n + n a − a ) (cid:17) . .1. THE TRANSVECTION QUADRUPLE OF AN ELEMENTARY NORMAL j -GROUP 107 Proof.
By definition we have s q q ′ = q ˜ σ (cid:0) q − q ′ (cid:1) Q and the formula for thesymmetry follows easily from (6.6) and (6.9). The formula for the midpoint mapcomes from a direct computation of the inverse diffeomorphism of the partial map s q := [ Q ∋ q ′ s q ′ q ∈ Q ]. (cid:3) Remark . Setting A := exp { a } and N := exp { l ⋆ ⊕ } we have the globaldecomposition Q = AN and for q = an ∈ Q , the symmetry at the neutral elementreads s e q = a − n − . Also, the global chart(6.12) e G/B → R d +1 , ( a, n ⊕ m , n ⊕ m , t , t , ℓ ) B ( a, n − n ) ,a, t , t , ℓ ∈ R , n , n ∈ l ⋆ , m , m ∈ l , identifies ˜ G/B with Q via the coordinatesystem (6.10). Lemma . The Abelian subgroup Y := exp { Y } of ˜ G , endows the local sym-plectic symmetric space ( S , s, ω S ) with an elementary structure in the sense of Def-inition 5.35. Proof.
We already know by Lemma 6.8, that the left invariant symmetricspace (
Q, s ) admits a midpoint map. We also know by Lemma 6.5 that Y isnormalized by Q and that S is isomorphic to Q ⋉ Y . But we need to knowthat Q ⋉ Y acts simply transitively on the symmetric space ˜ G/ ˜ K . For this, let g = ( a, v, , t, , t ) ∈ Q ⋉ Y and g ′ = ( a ′ , v ′ , v ′ , t ′ , t ′ , ℓ ′ ) ∈ ˜ G . Then we get gg ′ = (cid:0) a + a ′ , e − a ′ v + v ′ , v ′ , e − a ′ t + t ′ + e − a ′ ω ( v, v ′ ) , t ′ ,ℓ ′ + e − a ′ t + ω ( e − a ′ v, v ′ − v ′ ) (cid:1) , and thus in the chart (6.7) of ˜ G/ ˜ K , we get: gg ′ ˜ K (cid:0) a + a ′ , e − a ′ v + v ′ − v ′ , e − a ′ t + t ′ − t ′ − ω ( v ′ , v ′ ) + ω ( e − a ′ v, v ′ − v ′ ) (cid:1) . This means that under the identification S ≃ ˜ G/ ˜ K , Q ⋉ Y ≃ S acts by left trans-lations and the second condition of Definition (5.35) is verified. For the thirdcondition, note that under the parametrization (6.5) of ˜ G , we have Y = (cid:8) (0 , m, , t, , t ) : m ∈ l , t ∈ R (cid:9) . Take q = ( a, n, , , , ∈ Q , a ∈ R , n ∈ l ⋆ and b = e y = (0 , m, , t, , t ) ∈ Y , m ∈ l , t ∈ R . A computation then shows that h ξ, (cid:0) Ad q − − Ad ( s e q ) − (cid:1) y i = 2 t sinh 2 a + 2 cosh a ω ( n, m ) , which entails that Ψ( a, n ) = (cid:0) a, n cosh a (cid:1) . (6.13)The last condition follows from (6.4). (cid:3) Remark . Parametrizing ˜ G as in (6.5), S ≃ ˜ G/ ˜ K as in (6.7) and Q ≃ ˜ G/B as in (6.12), we have the following expression for the action of ˜ G on S :( a, v , v , t , t , ℓ ) . ( a ′ , v ′ , t ′ )= (cid:0) a + a ′ , e − a ′ v − e a ′ v + v ′ , e − a ′ t − e a ′ t + t ′ − ω ( v , v ) (cid:1) , and on Q :( a, n ⊕ m , n ⊕ m , t , t , ℓ ) . ( a ′ , n ′ ) = (cid:0) a + a ′ , e − a ′ n − e a ′ n + n ′ (cid:1) .
08 6. QUANTIZATION OF K¨AHLERIAN LIE GROUPS
Remark . From similar methods than those leading to Lemma 2.27, wededuce that we have: d Q ≍ [( a, n ) ∈ Q cosh a + | n | (1 + e a )] . From the Remark above and in analogy with Remark 2.36, we define the Fr´echetvalued Schwartz space of Q , denoted S ( Q, E ), as the set of smooth functions suchthat all left (or right) derivatives decrease faster than any power of d Q . The latterspace is Fr´echet for the semi-norms: f ∈ S ( Q, E ) sup X ∈ U k ( q ) sup x ∈ Q n d Q ( x ) n (cid:13)(cid:13) e X f ( x ) (cid:13)(cid:13) j | X | k o , j, k, n ∈ N , or even for f ∈ S ( Q, E ) sup X ∈ U k ( q ) sup x ∈ Q n d Q ( x ) n (cid:13)(cid:13) X f ( x ) (cid:13)(cid:13) j | X | k o , j, k, n ∈ N . In this section, we specialize the different ingredients of our quantization mapin the case of the elementary symplectic symmetric space ( S , s, ω S ) determined byan elementary normal j -group. We also (re)introduce a real parameter θ in thedefinition of the character (5.7): χ θ ( b ) := exp { iθ h ξ, log( b ) i} , b ∈ B , θ ∈ R ∗ , which is globally defined as B is exponential. By Lemma 5.17 and Remark 6.4, theHaar measure d S on S (respectively d Q on Q ) is invariant under both s ⋆e (respectively s ⋆e ) and ˜ G . Observe that under the parametrization (6.8) of the group B , we have χ θ ( b ) = exp { iθ ℓ } . Note that within the chart (6.10), any left-invariant Haar measured Q on Q is a multiple of the Lebesgue measure on q (these facts are transparent inEquations (2.5), (6.11) and in Remark 6.11). Also, within the chart (2.4), any leftinvariant Haar measure d S on S is a multiple of the Lebesgue measure on q ⋉ Y .By Remark 5.36 (iii), the restriction to S = Q ⋉ Y of the induced representation U χ θ (that we denote by U θ from now on) of ˜ G on L ( Q, d Q ) reads within the charts(2.4) on S and (6.10) on Q : U θ ( a, v, t ) ψ ( a , n )= exp n iθ (cid:0) e a − a ) t + ω ( e a − a n − n , e a − a m ) (cid:1)o ψ (cid:0) a − a, n − e a − a n (cid:1) , where ( a, v, t ) ∈ S with a, t ∈ R and v = n ⊕ m ∈ l ⋆ ⊕ l = V and ( a , n ) ∈ Q with a ∈ R and n ∈ l ⋆ . Remark . In accordance with the notations of earlier chapters, from nowon, we make explicit the dependence on the parameter θ ∈ R ∗ in all the objectswe are considering. For instance, we now set Ω θ, m instead Ω m for the quantizationmap, ⋆ θ, m instead of ⋆ m for the associated composition product, K θ, m instead of K m for its three-points kernel, E θ instead of E for the one-point phase etc. Lemma . Within the coordinates (2.4) on S and under the decomposition v = n ⊕ m ∈ l ⋆ ⊕ l = V , the one-point phase E θ of Lemma 5.32 reads E θ ( a, v, t ) = exp (cid:8) − iθ (cid:0) t sinh 2 a + ω ( n, m ) cosh a (cid:1)(cid:9) . .2. QUANTIZATION OF ELEMENTARY NORMAL j -GROUPS 109 Proof.
Recall that from Lemma 5.37, we have: E θ ( q − e y ) = exp (cid:8) iθ h ξ, [Ψ( q ) , y ] i (cid:9) . Which from (6.13) becomes: E θ ( q − e y ) = exp (cid:8) iθ (cid:0) t sinh 2 a + ω ( n, m ) cosh a (cid:1) (cid:9) , with q = ( a, n, , , , ∈ Q , a ∈ R , n ∈ l ⋆ and b = e y = (0 , m, , t, , t ) ∈ Y , m ∈ l , t ∈ R . Since q − = ( − a, − e a n, , , , E θ ( qb ) = exp (cid:8) − iθ (cid:0) t sinh 2 a + ω ( n, m ) e a cosh a (cid:1) (cid:9) . One concludes by observing that the coordinates qb are related to the coordinates(2.4) through( a, v, t ) = ( a, n, , , , . (0 , m, , t − ω ( n, m ) , , t − ω ( n, m )) . (cid:3) From (6.11) and (6.13), we observe: (cid:12)(cid:12)
Jac s e ( a, n ) (cid:12)(cid:12) = 2 d +1 cosh d a , | Jac Ψ ( a, n ) | = 2 d +2 cosh 2 a cosh d a , so that the element m given in (5.12) reads: m ( a, n ) = 2 d +2 cosh / a cosh d a . From this, we deduce:
Proposition . Parametrizing S as in (2.4) and Q as in (6.10) , we have thefollowing expression for the action on L ( Q, d Q ) of the unitary quantizer Ω θ, m ( x ) , x ∈ S , associated with the polarized symplectic symmetric space underlying an ele-mentary normal j -group S : Ω θ, m ( a, v, t ) ψ ( a , n ) = 2 d +1 cosh(2 a − a ) / cosh( a − a ) d × exp n iθ (cid:0) sinh(2 a − a ) t + ω (cosh( a − a ) n − n , cosh( a − a ) m ) (cid:1)o × ψ (cid:0) a − a , a − a ) n − n (cid:1) . Remark . Observe that (cid:12)(cid:12)
Jac s e (cid:12)(cid:12) is s e -invariant, as it is an even functionof the variable a only. Thus, by Lemma 5.28 and Remark 5.42, we deduce thatthe unitary quantization map Ω θ, m is also compatible with the natural involutionsof its source and range spaces (the complex conjugation on L ( S , d S ) and the ad-joint on L ( L ( Q, d Q )). In particular, it sends real-valued functions to self-adjointoperators.Our next result is one of the key steps of this chapter: it renders transparent thelink between chapters 3 and 4 and chapters 5 and 6. For this, we need the explicitexpression of the tri-kernel K θ, m of the product (5.14) for an elementary normal j -group S . First, observe that the unique solution of the equation s q ◦ s q ◦ s q ( q ) = q , q, q , q , q ∈ Q , as given in Lemma 5.53, reads q = (cid:0) a − a + a , cosh( a − a ) n − cosh( a − a ) n + cosh( a − a ) n (cid:1) . From [ , ], we extract Lemma . Within the notations of Proposition 5.55, we have J = (cid:12)(cid:12) Jac Φ Q (cid:12)(cid:12) .
10 6. QUANTIZATION OF K¨AHLERIAN LIE GROUPS
Then, Proposition 5.55 and a straightforward computation gives K θ, m = A m e iθ S with: A m ( x , x , x ) = m ( a − a ) m ( a − a ) m ( a − a )= 2 d +3 cosh(2 a − a ) / cosh( a − a ) d cosh(2 a − a ) / × cosh( a − a ) d cosh(2 a − a ) / cosh( a − a ) d , and S ( x , x , x ) = sinh(2 a − a ) t + sinh(2 a − a ) t + sinh(2 a − a ) t + cosh( a − a ) cosh( a − a ) ω ( v , v )+ cosh( a − a ) cosh( a − a ) ω ( v , v )+ cosh( a − a ) cosh( a − a ) ω ( v , v ) . By identification, we thereby obtain:
Proposition . For g , g , g ∈ S and θ ∈ R ∗ , we have K θ, m ( g , g , g ) = K θ, ( g − g , g − g ) , where the three-point kernel on the left hand side of the above equality is given inProposition 5.55 and the two-point kernel on the right hand side is given in Theorem3.5 for τ = 0 . In particular, the products ⋆ θ, m and ⋆ θ, coincide on L ( S , d S ) . Before giving the link between the generic kernels K θ, m and K θ,τ , hence afortiori between the generic products ⋆ θ, m and ⋆ θ,τ , we will give the relation betweenour quantization map and Weyl’s. Denote by Ω , the Weyl quantization map of S in the Darboux chart (2.4). For a function f on S , Ω ( f ) is an operator on L ( R d +1 ) ≃ L ( Q, d Q ) given (up to a normalization constant) byΩ ( f ) ψ ( a , n ) := C ( θ, d ) Z R d +2 e − iθ (2( a − a ) t + ω ( n − n,m )) f (cid:0) a + a , n + n , m, t (cid:1) ψ ( a, n ) d a d n d m d t . Then, recall that for τ ∈ Θ (see Definition 3.2), the inverse T − θ,τ of the map (3.1)is continuous on the ‘flat’ Schwartz space S ( S ). As the Weyl quantization mapscontinuously Schwartz functions to trace-class operators, we deduce that Ω ◦ T − θ,τ is well defined and continuous from S ( S ) to L (cid:0) L ( Q, d Q ) (cid:1) . From this, we get Proposition . Let τ ∈ Θ (see Definition 3.2). Then, as continuous oper-ators from S ( S ) to the trace ideal L (cid:0) L ( Q, d Q ) (cid:1) , we have (6.14) Ω ◦ T − θ,τ = Ω θ, m , where m ( a, n ) = m ( a, n ) exp (cid:8) τ (cid:0) θ sinh 2 a (cid:1)(cid:9) . Proof.
By density, it suffices to show that for f ∈ S ( S ), the operators Ω θ, m ( f )and Ω (cid:0) T ∗ θ,τ ( f ) (cid:1) coincide on D ( Q ). Note then that for f ∈ S ( S ), we have T − θ,τ ( f )( a, v, t ) =2 π Z R cosh / (cid:0) θξ (cid:1) cosh d (cid:0) θξ (cid:1) e τ (cid:0) θ sinh (cid:0) θξ (cid:1)(cid:1) f (cid:16) a, sech (cid:0) θξ (cid:1) v, t ′ (cid:17) e iξt − i θ sinh (cid:0) θξ (cid:1) t ′ d ξ dt ′ . By this we mean the ordinary Schwartz space in the global chart (2.4). Observe that S S can ( S ) ⊂ S ( S ). .2. QUANTIZATION OF ELEMENTARY NORMAL j -GROUPS 111 Hence, for ψ ∈ D ( Q ), we getΩ (cid:0) T − θ,τ ( f ) (cid:1) ψ ( a , n ) = C ( d ) Z R d +4 cosh / (cid:0) θξ (cid:1) cosh d (cid:0) θξ (cid:1) e τ (cid:0) θ sinh (cid:0) θξ (cid:1)(cid:1) e − iθ (2( a − a ) t + ω ( n − n,m )) e iξt − i θ sinh (cid:0) θξ (cid:1) t ′ × f (cid:16) a + a , sech (cid:0) θξ (cid:1) ( n + n ) / , sech (cid:0) θξ (cid:1) m, t ′ (cid:17) ψ ( a, n ) d ξ dt ′ d a d n d m d t . Performing the change of variable a a − a , n θξ ) n − n and m cosh( θξ ) m , we getΩ (cid:0) T − θ,τ ( f ) (cid:1) ψ ( a , n ) = C ( d ) Z R d +4 cosh / (cid:0) θξ (cid:1) cosh d (cid:0) θξ (cid:1) e τ (cid:0) θ sinh (cid:0) θξ (cid:1)(cid:1) f ( a, n, m, t ′ ) e iξt − i θ sinh (cid:0) θξ (cid:1) t ′ × e − iθ (cid:0) a − a ) t + ω (cid:0) n − cosh (cid:0) θξ (cid:1) n, cosh (cid:0) θξ (cid:1) m (cid:1)(cid:1) ψ (cid:0) a − a , θξ ) n − n (cid:1) × d ξ dt ′ d a d n d m d t . Integrating out the t -variable yields a factor δ (cid:0) ξ − θ ( a − a ) (cid:1) and thus we getΩ (cid:0) T − θ,τ ( f ) (cid:1) ψ ( a , n ) = C ( d ) Z R d +2 cosh / a − a ) cosh d ( a − a ) e τ (cid:0) θ sinh (cid:0) a − a ) (cid:1)(cid:1) f ( a, n, m, t ′ ) × e iθ (cid:0) sinh (cid:0) a − a ) (cid:1) t ′ + ω (cid:0) cosh( a − a ) n − n , cosh( a − a ) m (cid:1)(cid:1) × ψ (cid:0) a − a , a − a ) n − n (cid:1) d a d n d m d t ′ , which by Proposition 6.15 coincides with Ω θ, m ( f ) ψ . (cid:3) From the above result and the defining relation (3.2) for the product ⋆ θ,τ for ageneric element τ ∈ Θ , we then deduce: Proposition . To every τ ∈ Θ , associate a right- N -invariant function m on Q as in (6.14) . Then, the three point kernel K θ, m of the product ⋆ θ, m definedin Proposition 5.52 (ii), is related to the two-point kernel K θ,τ given in Theorem3.5 via: K θ, m ( g , g , g ) = K θ,τ ( g − g , g − g ) , ∀ g , g , g ∈ S . In particular, the product ⋆ θ, m is well defined on B ( S ) and coincide with ⋆ θ,τ . Remark . From now on, to indicate that a right- N -invariant borelianfunction m on Q is associated to an element τ ∈ Θ , as in (6.14), we just write m ∈ Θ ( S ). Remark . Observe that, considering the ‘medial triangle’ three-point func-tion Φ S given in Proposition 2.14 (ii), and writing K θ, m = A m e iθ S , for m a right- N -invariant function on Q , we have: A m ( x , x , x ) = m ( a − a ) m ( a − a ) m ( a − a ) m ( a − a )= (cid:12)(cid:12) Jac Φ − S ( x , x , x ) (cid:12)(cid:12) / mm ( a − a ) m m ( a − a ) mm ( a − a ) . See Remark 6.9 for the definition of the abelian subgroup N .
12 6. QUANTIZATION OF K¨AHLERIAN LIE GROUPS
The expression above for the amplitude (in the case where m is right- N -invariant),could also be derived from the explicit expression for the Berezin transform (seeDefinition 5.46). Indeed, in the present situation, its distributional kernel in coor-dinates (2.4), reads: B θ, m [ x , x ]= δ ( a − a ) δ ( n − n ) δ ( m − m ) Z R | m | m (cid:0) arcsinh( a ) (cid:1) e iθ a ( t − t ) d a . By standard Fourier-analysis arguments, we deduce: B θ, m = | m | m (cid:0) arcsinh( iθ ∂ t ) (cid:1) . Finally, let us discuss the question of the involution for the generic product ⋆ θ, m . Since in general, the formal adjoint of Ω θ, m ( x ) on L ( Q, d Q ) is Ω θ,s ⋆e m ( x ),we deduce f ⋆ θ, m f = f ⋆ θ,s ⋆e m f . Hence, we obtain that the natural involution for the product ⋆ θ, m is(6.15) ∗ θ, m : f m s ⋆e m (cid:0) arcsinh( iθ ∂ t ) (cid:1) f . Remark . From Theorem 5.43 and the previous expression for the involu-tion, we observe that the element m defined in (5.12) is uniquely determined by therequirement that the associated quantization map is both unitary and involutionpreserving. Consider now a normal j -group B = ( S N ⋉ R N − . . . ) ⋉ R S with associatedextension morphisms R j ∈ Hom (cid:0) ( S N ⋉ . . . ) ⋉ S j +1 , Sp( V j , ω j ) (cid:1) , j = 1 , . . . , N − , (6.16)as in (2.3). We wish to apply Proposition 5.62 to this situation. For this, recall thatfor S an elementary normal j -group viewed as an elementary symplectic symmetricspace, D denotes the Lie algebra of W -preserving symplectic endomorphisms of p where the Lagrangian subspace W has been chosen to be l − ⊕ R E − ≃ l ⊕ R E . Proposition . Denote by D the stabilizer Lie subalgebra in sp ( V, ω ) ofthe Lagrangian subspace l . (i) Let
Sym( l ) be the space of endomorphisms of l that are symmetric withrespect to a given Euclidean scalar product on l . Let also η : End ( l ) × Sym( l ) → Sym( l ) , ( T, S ) T ◦ S + S ◦ T t . Then, endowing
Sym( l ) with the structure of an Abelian Lie algebra, onehas the isomorphism: D ≃ End ( l ) ⋉ η Sym( l ) . (ii) The Lie algebra D contains an Iwasawa component of sp ( V, ω ) . (iii) Letting D trivially act on the central element E of h induces an isomor-phism: D ≃ D ⋉ h . .3. QUANTIZATION OF NORMAL j -GROUPS 113 Proof.
Item (i) is immediate from an investigation at the matrix form level.Item (iii) follows from the fact that the derivation algebra of the non-exactpolarized transvection symplectic triple ( g , σ, ̟ ) underlying an elementary normal j -group S admits the symplectic Lie algebra sp ( V, ω ) as Levi-factor [ ].Item (ii) follows from a dimensional argument combined with Borel’s conju-gacy Theorem of maximal solvable subgroups in complex simple Lie groups. In-deed, on the first hand, the dimension of the Iwasawa factor of sp ( V, ω ) equalsdim sp ( V, ω ) − dim u ( d ) that is 2 d + d (2 d − − d = d ( d + 1) with 2 d := dim V =2 dim l . On the other hand, the dimension of the Borel factor in End ( l ) equals d + d ( d − which equals dim Sym( l ). Hence D contains a maximal solvable Liesubalgebra of dimension 2( d + d ( d − ) = d ( d + 1). Borel’s Theorem then yields theassertion since sp ( V, ω ) is totally split. (cid:3) From [ ], we observe that the full polarization quadruple of S underlies the Liegroup L = Sp( V, ω ) ⋉ ˜ G . Hence: Corollary . Let S be an elementary normal j -group viewed as an ele-mentary symplectic symmetric space (see Definition 5.35) and let ( L , σ L , ξ, B ) bethe associated full polarization quadruple (see Definition 5.19). Then we have theglobal decomposition L = Q B and moreover ∆ L (cid:12)(cid:12) B = ∆ B . We can now prove the conditions needed to apply Proposition 5.62:
Proposition . Let B = ( S N ⋉ R N − . . . ) ⋉ R S be a normal j -group, towhich one associates the full polarization quadruples ( L j , σ L j , ξ j , B j ) , j = 1 , . . . , N ,of the S j ’s. Then, there exists an homomorphism ρ j : ( S N ⋉ . . . ) ⋉S j → ˜ D j − whoseimage normalizes S j − in L j − and such that the extension homomorphism R j constructed in (5.16) , coincides with the extension homomorphism R j underlyingthe Pyatetskii-Shapiro’s decomposition (6.16) . Proof.
Firstly, by Pyatetskii-Shapiro’s theory [ ], one knows that the actionof ( S N ⋉ . . . ) ⋉ S j on S j − factors through a solvable subgroup of Sp( V j − , ω j − ).Setting ( AN ) j − the Iwasawa factor of Sp( V j − , ω j − ), we thus get an homomor-phism: ˜ ρ j : ( S N ⋉ . . . ) ⋉ S j → ( AN ) j − . But Proposition 6.24 (ii) asserts that ( AN ) j − is a subgroup of exp { D ,j − } , where D ,j − is the stabilizer Lie subalgebra in sp ( V j − , ω j − ) of the Lagrangian subspace l j − . Combining this with the isomorphism of Proposition 6.24 (iii), yields anotherhomomorphism: ˆ ρ j : ( AN ) j − → D j − ⊂ ˜ D j − . Hence ρ j := ˆ ρ j ◦ ˜ ρ j is the desired homomorphism. Now, observe that by [ , Propo-sition 2.2 item (i)], the group S j − viewed as a subgroup of L j − , is normalizedby Sp( V j − , ω j − ) for the action given in Proposition 2.9 (iv) and that this actionis precisely the one associated with the extension homomorphism R j in the de-composition (6.16). Thus, all that remains to do is to prove that the extensionhomomorphisms R j and R j coincide. Here, R j ( g ) := C ρ j ( g ) ∈ Aut ( S j − ), g ∈ S j ,is the extension homomorphism constructed in (5.16). But that R j = R j fol-lows from a very general fact about homogeneous spaces. Namely, observe that if M = G/K , then action of the isotropy K × M → M , ( k, gK ) → kgK , lifts to G as
14 6. QUANTIZATION OF K¨AHLERIAN LIE GROUPS the restriction to K of the conjugacy action K × G → G , ( k, g ) kgk − (indeed: kgK = kgk − K ). (cid:3) From this, we deduce that Proposition 5.62 and Theorem 5.63 are valid in thecase of a normal j -group. Moreover, we also deduce that the associated product ⋆ θ, m coincides with ⋆ θ,~τ of Proposition 3.7: Proposition . Let B be a normal j -group. To every ~τ ∈ Θ N , we associatea function m on Q N × . . . Q by m = m N ⊗ · · · ⊗ m where m j is related to τ j as in (6.14) . Then, the three-point kernel K θ, m of the product ⋆ θ, m defined in Theorem5.63 (iii), is related with the two-point kernel K θ,τ given in Proposition 3.7 (3.5) via: K θ, m ( g , g , g ) = K θ,~τ ( g − g , g − g ) , ∀ g , g , g ∈ B . In particular, the product ⋆ θ, m is well defined on B ( B ) and coincide with ⋆ θ,~τ . Remark . To indicate that a function m = m N ⊗ · · · ⊗ m on Q N × . . . Q is related to elements τ j ∈ Θ as in (6.14), we just write m ∈ Θ ( B ).We also quote the following extension of Proposition 6.19: Proposition . Let B be a normal j -group. For any m ∈ Θ ( B ) , the quan-tization map Ω θ, m is a continuous operator from S ( B ) := S ( S N ) ⊗ · · · ⊗ S ( S ) to L (cid:0) L ( Q N , d Q N ) ⊗ · · · ⊗ L ( Q , d Q ) (cid:1) . Our last result concerns the representation homomorphism (5.17).
Lemma . Let B = B ′ ⋉ R S , S = Q ⋉ Y . Then, the restriction of thehomomorphism R : B ′ → U ( L ( Q )) (see (5.17) ) underlying the one ρ =: B ′ → ˜ D of Proposition 6.26, defines a tempered action (see Definition 4.1) of B ′ on S ( Q ) . Proof.
Parametrizing S ≃ s = { ( a, n, m, t ) } , Q ≃ q = { ( a, n ) } as usual,the matrix of the Pyatetskii-Shapiro extension homomorphism R g ′ is expressedunder the form R g ′ = R + ( g ′ ) 0 00 R − ( g ′ ) ( R + ( g ′ ) T ) −
00 0 0 0 , where (by Borel’s conjugacy theorem) the element R + ( g ′ ) is (conjugated to) anupper triangular matrix form in End ( l ⋆ ). The determinant of this element thenconsists (as a small induction argument shows) in a product of powers of the mod-ular functions of the elementary factors of S , ..., S N of B ′ :det( R + ) = Π Nj =2 ∆ n j S j . One deduces from a careful examination of the construction of ρ in Proposition6.26, that R is explicitly given by: R ( g ′ ) ϕ ( a, n ) = det( R + ( g ′ )) − ϕ ( a, R + ( g ′ ) − n ) . Temperedeness results in a direct computation using that the semi-norms of S ( Q )are given by (see the discussion right after Remark 6.12): k f k k,n := sup X ∈ U k ( q ) sup x ∈ Q n d Q ( x ) n (cid:12)(cid:12) e X f ( x ) (cid:12)(cid:12) j | X | k o , .3. QUANTIZATION OF NORMAL j -GROUPS 115 that left invariant vector fields associated with H the generator of a and { f j } dj =1 be a basis of l ⋆ , read in the coordinates (6.10): e H = ∂ a − d X j =1 n j ∂ n j , e f j = ∂ n j , j = 1 , . . . , d , together with the behavior of the modular weight d Q given in Remark 6.12). (cid:3) HAPTER 7
Deformation of C ∗ -algebras Throughout this chapter, we consider a C ∗ -algebra A , endowed with an iso-metric and strongly continuous action α of a normal j -group B . In chapter 4, wehave seen how to deform the Fr´echet algebra A ∞ , consisting of smooth vectors forthe action α . Our goal here is to construct a C ∗ -norm on ( A ∞ , ⋆ αθ, m ), in order toget, after completion, a deformation theory at the C ∗ -level. We stress that fromnow on, the isometricity assumption of the action is fundamental. The way we willdefine this C ∗ -norm is based on the pseudo-differential calculus introduced in theprevious two chapters.The basic ideas of the construction can be summarized as follow. Consider H ,a separable Hilbert space carrying a faithful representation of A . We will thereofidentify A with its image in B ( H ). Let S S B can ( B , A ) be the A -valued one-pointSchwartz space associated to the tempered pair ( B × B , S B can ) as given in Defini-tion 1.40. Since this space is a subset of the flat A -valued Schwartz space of B ,Proposition 6.29 shows that for every m ∈ Θ ( B ), the map(7.1) f ∈ S S B can ( B , A ) Ω B θ, m ( f ) := Z B Ω B θ, m ( x ) ⊗ f ( x ) d B ( x ) , is well defined and takes values in K (cid:0) H χ (cid:1) ⊗ A , where H χ := L ( Q ) ⊗· · ·⊗ L ( Q N ).Then, the main step is to extend the map (7.1), from S S B can ( B , A ) to B ( B , A ). As for a ∈ A ∞ , the A -valued function α ( a ) := [ g ∈ B α g ( a ) ∈ A ] belongs to B ( B , A ),we will define a new norm on A ∞ by setting k a k θ, m := (cid:13)(cid:13) Ω B θ, m (cid:0) α ( a ) (cid:1)(cid:13)(cid:13) , where the norm on the right hand side above, denotes the C ∗ -norm of B (cid:0) H χ ⊗ H (cid:1) .This will eventually be achieved by proving a non-Abelian C ∗ -valued version of theCalder´on-Vaillancourt Theorem in the context of the present pseudo-differential cal-culus. This theorem will be proved using wavelet analysis and oscillatory integralsmethods. Let B be a normal j -group, with Pyatetskii-Shapiro decomposition B = ( S N ⋉ . . . ) ⋉ S , where the S j ’s, j = 1 , . . . , N , are elementary normal j -groups. Recall that ourchoice of parametrization is: S N × · · · × S → B , ( g N , . . . , g ) g . . . g N . Given a Hilbert space H , K ( H ) denotes the C ∗ -algebra of compact operators. C ∗ -ALGEBRAS Remark . Observe that the extension homomorphism at each step, R j ,being valued in Sp( V j , ω j ), it preserves any left invariant Haar measure d S j on S j :(7.2) ( R jg ′ ) ⋆ d S j = d S j , ∀ g ′ ∈ ( S N ⋉ . . . ) ⋉ S j − . This implies that the product of left invariant Haar measures d S ⊗ · · ·⊗ d S N definesa left invariant Haar measure on B under both parametrizations g = g . . . g N or g = g N . . . g of g ∈ B .The aim of this section is to construct a weak resolution of the identity on thetensor product Hilbert space L ( Q N ) ⊗ · · · ⊗ L ( Q ) := H χ from a suitable familyof coherent states for B , that we now introduce. Definition . Let B be a normal j -group. Given a mother wavelet η ∈D ( Q N × · · · × Q ), let { η x } x ∈ B be the family of coherent states defined by η x := U θ ( x ) η , x ∈ B , where U θ is the unitary representation of B on H χ constructed in Lemma 5.59 forthe morphism underlying Proposition 6.26.Observe that in the elementary case, we have: η x ( q ) = E θ ( q − q b ) η ( q − q ) , x = qb ∈ S , q ∈ Q , (7.3)where the phase E θ is defined by E θ ( x ) := χ θ (cid:0) C q − ( b ) (cid:1) , x = qb ∈ S . (7.4)In the generic case, setting B = B ′ ⋉ R S with B ′ a normal j -group and S anelementary normal j -group, for η = η ′ ⊗ η , η ′ ∈ D ( Q N × · · · × Q ), η ∈ D ( Q )and parametrizing g ∈ B as g = g ′ .g , g ′ ∈ B ′ , g ∈ S , we have (see Lemma 5.59): η g = η ′ g ′ ⊗ R ( g ′ ) η g , (7.5)where R : B ′ → U ( L ( Q )) is the homomorphism underlying (5.17). Parametrizingnow g ∈ B as g = g .g ′ , g ′ ∈ B ′ , g ∈ S , we find η g = η ′ g ′ ⊗ (cid:0) R ( g ′ ) η (cid:1) g , (7.6) Proposition . Let B be a normal j -group, E a complex Fr´echet space and η ∈ D ( Q N × · · · × Q ) . Then, the map F η : L ∞ ( Q N × · · · × Q , E ) → L ∞ ( B , E ) ,f h x ∈ B Z Q ×···× Q N f ( q N , . . . , q ) η x ( q N , . . . , q )d Q N ( q N ) . . . d Q ( q ) ∈ E i , restricts as a continuous map F η : S ( Q N × · · · × Q , E ) → S S can ( B , E ) . Proof.
Assume first that B = S is an elementary normal j -group. We denoteby E θ the element of C ∞ ( S ) given in (7.4), so that with x = qb ∈ S , q ∈ Q , b ∈ Y ,we have for every f ∈ S ( Q, E ): (cid:0) F η f (cid:1) ( x ) = Z Q f ( q ) E θ ( q − q b ) η ( q − q )d Q ( q ) = Z Q f ( qq ) E θ ( q b ) η ( q ) d Q ( q ) . Decomposing as usual q = a ⊕ n , we let H be the generator of a and { f j } dj =1 bea basis of n . Then, from the expressions given in (2.8), we see that the associatedleft invariant vector fields read in the coordinates (6.10): .1. WAVELET ANALYSIS 119 e H = ∂ a − d X j =1 n j ∂ n j , e f j = ∂ n j , j = 1 , . . . , d . Moreover, in the chart (2.4) of S , with x = ( a, n ⊕ m, t ), the function E θ takes thefollowing form: E θ ( x ) = exp (cid:8) iθ (cid:0) e − a t − e − a ω ( n, m ) (cid:1)(cid:9) . Hence, defining i e H E θ =: α E θ and − d X j =1 e f j E θ =: β E θ , a simple computation gives α ( x ) = θ ( e − a t − e − a ω ( n, m )) , β ( x ) = θ − e − a | m | , where | m | = P dj =1 ω ( f j , m ) . Moreover, it is easy to see that both α and β are eigenvectors of e H with eigenvalue − e f j β = 0. Hence setting e P :=1 − P dj =1 e f j , we get by integration by parts on the q -variable and with k, k ′ ∈ N arbitrary: (cid:0) F η f (cid:1) ( x ) = Z Q E θ ( q b )(1 − e H k ′ q ) " e P kq (cid:2) f ( qq ) η ( q ) (cid:3)(cid:0) α ( q b ) k ′ + α ( k ′ ) ( q b ) (cid:1)(cid:0) β ( q b ) (cid:1) k d Q ( q )where α ( k ′ ) := k ′ − X r =1 c k ′ r α r ( c k ′ r ∈ C ) . This easily entails that (cid:13)(cid:13)(cid:0) F η f (cid:1) ( x ) (cid:13)(cid:13) j ≤ C ( k, k ′ ) Z Q k (1 − e H k ′ q ) e P kq [ f ( qq ) η ( q )] k j (cid:0) α ( q b ) k ′ (cid:1)(cid:0) β ( q b ) (cid:1) k d Q ( q ) . By left invariance of e H and e P , we get up to a redefinition of f ∈ S ( Q, E ) and of η ∈ D ( Q ): k (cid:0) F η f (cid:1) ( x ) k j ≤ C ( k, k ′ ) Z Q k f ( qq ) k j | η ( q ) | (cid:0) α ( q b ) k ′ (cid:1)(cid:0) β ( q b ) (cid:1) k d Q ( q ) . Now, given any tempered weight µ on S , one therefore has Z S µ ( x ) k (cid:0) F η f (cid:1) ( x ) k j d S ( x ) ≤ C ( k, k ′ ) Z Q × Y µ ( qb ) k f ( qq ) k j | η ( q ) | (cid:0) α ( q b ) k ′ (cid:1)(cid:0) β ( q b ) (cid:1) k d Q ( q )d Q ( q ) d Y ( b ) . Observing that for every X ∈ U ( q ), the element e X q [ qb E θ ( qb )] only dependson the variable C q − ( b ) =: exp Ad q − y , where b = e y ∈ Y , changing the variable
20 7. DEFORMATION OF C ∗ -ALGEBRAS following b :=: exp( y ) := C q − ( b ) ( y ∈ Y ) yields Z Q × Y µ ( qb ) k f ( qq ) k j | η ( q ) | (cid:0) α ( q b ) k ′ (cid:1)(cid:0) β ( q b ) (cid:1) k d Q ( q )d Q ( q ) d Y ( b ) = Z Q × Y µ ( qq b q − ) | det Ad q | Y | k f ( qq ) k j | η ( q ) | (cid:0) α ( b ) k ′ (cid:1)(cid:0) β ( b ) (cid:1) k d Q ( q )d Q ( q ) d Q ( b ) . Writing y = ( m , t ) ∈ l × R as before, we observe that α ( b ) = t θ , β ( b ) = θ − | m | and d Y ( b ) = d y . Therefore this last integral exists as soon as k and k ′ are large enough in front of the polynomial growth of µ . Since for every Y ∈ U ( s ),one has e Y x [ x η x ] = ( dU θ ( Y ) η ) x , the left-invariant derivatives of F η f follow theexact same treatment.Passing to non-elementary case, we first observe that by Lemma 6.30, the restrictedaction R of the tempered Lie group B ′ on S ( Q ) is tempered in the sense of Defini-tion 4.1. Therefore, in the above discussion, replacing η by R ( g ′ ) η yields constants C ( k, k ′ ) that now depend polynomially on the tempered weights { µ R k,j ∈ C ∞ ( B ′ ) } associated to the tempered action R (within the notations of Definition 4.1).Now, within the conventions of (7.5) and considering f := f ′ ⊗ f ∈ S ( Q N × ... × Q , E ) ⊗S ( Q ) and η := η ′ ⊗ η ∈ D ( Q N × ... × Q ) ⊗D ( Q ), one has:( F η f ) ( g g ′ ) = (cid:0) F η ′ f ′ (cid:1) ( g ′ ) (cid:0) F R ( g ′ ) η f (cid:1) ( g ) . Let d R the infinitesimal form of R , that is for η ∈ D ( Q ), it is defined by d R ( X ) η := ddt | t =0 R ( e tX ) η for X ∈ b ′ and extended as an algebra morphismto U ( b ′ ). Note that for any X ∈ U ( b ′ ) and g ′ ∈ B ′ , we have e X g ′ R ( g ′ ) η = R ( g ′ ) (cid:0) d R ( X ) η (cid:1) . Now, for every tempered weight µ = µ ′ ⊗ µ on B and X = X ′ ⊗ X ∈ U ( b ) (obviousnotations), denoting ˜ η := d R ( X ′ (2) ) η ∈ D ( Q N × · · · × Q ) , we then have Z B µ ( g ) k e X. ( F η f ) ( g ) k j d B ( g )(7.7) ≤ X ( X ′ ) Z B ′ µ ′ ( g ′ ) Z S µ ( g ) k g X ′ (1) g ′ (cid:0) F η ′ f ′ ( g ′ ) (cid:1) g X ′ (2) g ′ f X g (cid:0) F R ( g ′ ) η f ( g ) (cid:1) k j d g d g ′ = X ( X ′ ) Z B ′ µ ′ ( g ′ ) k g X ′ (1) g ′ (cid:0) F η ′ f ′ ( g ′ ) (cid:1) k j Z S µ ( g ) (cid:12)(cid:12) f X g (cid:0) F R ( g ′ )˜ η ) f ( g ) (cid:1)(cid:12)(cid:12) d g d g ′ From what we have proven in the elementary case and by induction hypothesis on N , the map B ′ → R + , g ′ Z S µ ( g ) (cid:12)(cid:12) f X g (cid:0) F R ( g ′ )˜ η ) f ( g ) (cid:1)(cid:12)(cid:12) d g , is temperate. Hence, the integral (7.7) exists. We conclude by nuclearity of theSchwartz spaces S ( Q N × · · · × Q , E ). (cid:3) We then deduce the following consequence: .1. WAVELET ANALYSIS 121
Corollary . Let B be a normal j -group and η ∈ D ( Q N × · · · × Q ) . Then,the maps [ B ∋ x
7→ h η, η x i ] and [ B ∋ x
7→ h η, η x − i ] belong to L ( B ) . Proof.
This follows from Proposition 7.3 with E = C since h η, η x i = F η (cid:0) η (cid:1) ( x )and h η, η x − i = h η, η x i . (cid:3) The next result is probably well known but since we are unable to locate it inthe literature and since we use it several times, we deliver a proof. Lemma . Let ( X, µ ) be a σ -finite measure space and H a separable Hilbertspace. Consider an element K ∈ L ∞ (cid:0) X × X, µ ⊗ µ ; B ( H ) (cid:1) , such that c := sup x ∈ X Z X (cid:13)(cid:13) K ( x, y ) (cid:13)(cid:13) B ( H ) dµ ( y ) < ∞ , c := sup y ∈ X Z X (cid:13)(cid:13) K ( x, y ) (cid:13)(cid:13) B ( H ) dµ ( x ) < ∞ . Then, the associated kernel operator is bounded on L ( X, µ ; H ) ≃ L ( X, µ ) ⊗ H with operator norm not exceeding c c . Proof.
Let T K be the operator associated with the kernel K . For vectorsΦ , Ψ ∈ L ( X, µ ; H ) ∩ L ( X, µ ; H ), we have (cid:12)(cid:12) h Φ , T K Ψ i (cid:12)(cid:12) < ∞ . Moreover, the Cauchy-Schwarz inequality gives (cid:12)(cid:12) h Φ , T K Ψ i (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z X × X (cid:10) Φ( x ) , K ( x, y ) Ψ( y ) (cid:11) H dµ ( y ) dµ ( x ) (cid:12)(cid:12)(cid:12) ≤ Z X × X k Φ( x ) k H k K ( x, y ) k B ( H ) k Ψ( y ) k H dµ ( y ) dµ ( x ) ≤ (cid:16) Z X × X k Φ( x ) k K k K ( x, y ) k B ( H ) dµ ( y ) dµ ( x ) (cid:17) / × (cid:16) Z X × X k K ( x, y ) k B ( H ) k Ψ( y ) k H dµ ( y ) dµ ( x ) (cid:17) / ≤ c c k Φ k L ( X,µ ; H ) k Ψ k L ( X,µ ; H ) , and the claim follows immediately. (cid:3) We are now able to prove that the family of coherent states { η x } x ∈ B providesa weak resolution of the identity. Proposition . Let B be a normal j -group, H a separable Hilbert space and η ∈ D ( Q N × · · · × Q ) \ { } . Then, for all Φ , Ψ ∈ H χ ⊗ H the following relationholds: h Ψ , Φ i H χ ⊗H = C B ( η ) − Z B (cid:10) h η x , Ψ i H χ , h η x , Φ i H χ (cid:11) H d B ( x ) , (7.8) where h η x , Ψ i H χ is the vector in H is defined by: (cid:10) ϕ, h η x , Ψ i H χ (cid:11) H := (cid:10) η x ⊗ ϕ, Ψ (cid:11) H χ ⊗H , ∀ ϕ ∈ H , and C B ( η ) := (2 πθ ) dim( B ) / k ∆ Q N ×···× Q η k , where ∆ Q N ×···× Q = ∆ Q N ⊗ · · · ⊗ ∆ Q , with ∆ Q j the modular function of Q j . It can be viewed as a Banach space valued version of Schur’s test Lemma.
22 7. DEFORMATION OF C ∗ -ALGEBRAS Proof.
We first demonstrate that for Φ ∈ H χ ⊗ H , the map [ x ∈ B η x , Φ i H χ ∈ H ] belongs to L ( B , H ). To see this, let { B j } j ∈ N be an increasingsequence of relatively compact subsets of B , which converges to B . For each j ∈ N ,we define the operator T ηj : L ( B j , H ) → H χ ⊗ H , F Z B j η x ⊗ F ( x ) d B ( x ) . Clearly, each T ηj is bounded: k T ηj F k H χ ⊗H ≤ Z B j k η x k H χ k F ( x ) k H d B ( x ) ≤ k η k H χ meas( B j ) / (cid:16) Z B j k F ( x ) k H d B ( x ) (cid:17) / = k η k H χ meas( B j ) / k F k L ( B , H ) . To see that the family { T ηj } j ∈ N is in fact uniformly bounded, note that the adjointof T ηj reads: T ηj ∗ : H χ ⊗ H → L ( B j , H ) , Φ (cid:2) x ∈ B j
7→ h η x , Φ i H χ ∈ H (cid:3) . Hence for F ∈ L ( B j , H ) we get (cid:12)(cid:12) T ηj (cid:12)(cid:12) F ( x ) = Z B j h η x , η y i F ( y ) d B ( y ) , that is | T ηj | = S ηj ⊗ Id H , where S ηj ∈ B ( L ( B j )) is a kernel operator with kernel K ηj ( x, y ) = h η x , η y i . Applying Lemma 7.5, a simple change of variable gives k S ηj k ≤k [ x
7→ h η, η x i ] k := C which is finite by Lemma 7.4 and of course, is uniform in j ∈ N . Finally, since Z B j kh η x , Φ i H χ k H d B ( x ) = k T ηj ∗ Φ k L ( B j , H ) ≤ C k Φ k H χ ⊗H , taking the limit j → ∞ gives Z B kh η x , Φ i H χ k H d B ( x ) ≤ C k Φ k H χ ⊗H , as needed. The rest of the proof is computational. Assume first that B = S iselementary. In this case, for Φ , Ψ ∈ H χ ⊗ H and η ∈ D ( Q ), we have in chart (2.4)and from (7.3): Z S (cid:10) h η x , Ψ i H χ , h η x , Φ i H χ (cid:11) H d S ( x ) = Z R d +4 h Ψ( a , n ) , Φ( a , n ) i H η (cid:0) a − a, n − e a − a n (cid:1) η (cid:0) a − a, n − e a − a n (cid:1) × exp n iθ (cid:0) ( e − a − e − a ) e a (cid:0) t + ω ( n, m ) (cid:1) − e a ω ( n e − a − n e − a , m ) (cid:1)o × d a d n d m d t d a d n d a d n . .1. WAVELET ANALYSIS 123 Integrating out the t -variable yields a factor 2 πθ e − a +2 a δ ( a − a ), the formerexpression then becomes2 πθ Z R d +2 h Ψ( a , n ) , Φ( a , n ) i H η (cid:0) a − a, n − e a − a n (cid:1) η (cid:0) a − a, n − e a − a n (cid:1) × e − a +2 a exp n − iθ (cid:0) e a − a ω ( n − n , m ) (cid:1)o d a d n d m d a d n d n . Integrating the m -variables, yields a factor (2 πθ ) d e − da + da δ ( n − n ) and we get,up to a constant: Z R d +2 h Ψ( a , n ) , Φ( a , n ) i H η (cid:0) a − a, n − e a − a n (cid:1) η (cid:0) a − a, n − e a − a n (cid:1) × e − ( d +2)( a − a ) d a d n d a d n , which after an affine change of variable and restoring the constants, gives(2 πθ ) d +1 h Ψ , Φ i H χ ⊗H Z R d +1 | η ( a, n ) | e (2 d +2) a d a d n , which is all we needed since ∆ Q ( a, n ) = e ( d +1) a .The case of a generic normal j -group B is treated with the same argument lines as inProposition 7.3: set B ′ ⋉S , with S elementary normal and assume that the relation(7.8) holds for B ′ . With the notations of (7.5), we have for all ϕ = ϕ ′ ⊗ ϕ , ψ = ψ ′ ⊗ ψ ∈ H χ and omitting the constant C B ( η ) − = C B ′ ( η ′ ) − C S ( η ) − ): Z B h ϕ, η x ih η x , ψ i d B ( x )= Z B ′ × S h ϕ ′ , η ′ g ′ ih ϕ , R ( g ′ ) η g ihR ( g ′ ) η g , ψ ih η ′ g ′ , ψ ′ i d B ′ ( g ′ ) d S ( g )= Z B ′ h ϕ ′ , η ′ g ′ i (cid:16) Z S hR ( g ′− ) ϕ , η g ih η g , R ( g ′− ) ψ i d S ( g ) (cid:17) h η ′ g ′ , ψ ′ i d B ′ ( g ′ )= Z B ′ h ϕ ′ , η ′ g ′ ihR ( g ′− ) ϕ , R ( g ′− ) ψ ih η ′ g ′ , ψ ′ i d B ′ ( g ′ )= h ϕ , ψ i Z B ′ h ϕ ′ , η ′ g ′ ih η ′ g ′ , ψ ′ i d g ′ = h ϕ , ψ ih ϕ ′ , ψ ′ i = h ϕ, ψ i . (cid:3) Remark . In the following, we will absorb the constant C ( η ) / of Propo-sition 7.6 in a redefinition of the mother wavelet η ∈ D ( Q N × · · · × Q ). Remark . Other types of weak resolution of the identity can be constructedin this setting. For instance, setting ˜ η x := Ω B m ( x ) η , where η is arbitrary in H χ , wehave from the unitarity of the quantization map Ω B m : h ψ, φ i = k η k − Z B h ψ, ˜ η x ih ˜ η x , φ i d B ( x ) , for all φ, ψ ∈ H χ . Similarly, let W ηx,y be the Wigner function on B , associated to apair of wavelets η x , η y : W ηx,y ( z ) := σ m (cid:2) | η x ih η y | (cid:3) ( z ) = h η y | Ω m ( z ) | η x i , x, y, z ∈ B .
24 7. DEFORMATION OF C ∗ -ALGEBRAS Then, these Wigner functions may be used to construct a weak resolution of theidentity on L ( B ): For all f , f ∈ L ( B ), we have h f , f i = k η k − Z B × B h f , W ηx,y i h W ηx,y , f i d B ( x ) d B ( y ) . Proposition . Let B be a normal j -group, H be a separable Hilbert space and A be a densely defined operator on H χ ⊗ H , whose domain contains the (algebraic)tensor product D ( Q N × · · · × Q ) ⊗ H . For x, y ∈ B and η ∈ D ( Q N × · · · × Q ) define the element h η x , Aη y i H χ of B ( H ) by means of the quadratic form H × H → C , ( φ, ψ )
7→ h η x ⊗ φ, A η y ⊗ ψ i H χ ⊗H . Assuming further that sup y ∈ B Z B kh η x , Aη y i H χ k B ( H ) d B ( x ) < ∞ , sup x ∈ B Z B kh η x , Aη y i H χ k B ( H ) d B ( y ) < ∞ , then A extends to a bounded operator on H χ ⊗ H . Proof.
Since η x is smooth and compactly supported, our assumption aboutthe domain of A ensures that h η x , Aη y i H χ is well defined as an element of B ( H ).Thus, Lemma 7.5 applied to ( X, µ ) = ( B , d B ), yields that the operator ˜ A on L ( B , H ) given by ˜ AF ( x ) := Z B h η x , Aη y i H χ F ( y ) dµ ( y ) , and is bounded, with k ˜ A k ≤ (cid:16) sup y ∈ B Z B kh η x , Aη y i H χ k B ( H ) d B ( x ) (cid:17) / × (cid:16) sup x ∈ B Z B kh η x , Aη y i H χ k B ( H ) d B ( y ) (cid:17) / < ∞ . For Φ ∈ H χ ⊗ H define the H -valued function on B : ˜Φ := [ x ∈ B
7→ h η x , Φ i H χ ∈H ]. By Proposition 7.6 we know that ˜Φ belongs to L ( B , H ) with k Φ k H χ ⊗H = k ˜Φ k L ( B , H ) . Take now Φ , Ψ ∈ dom A . In this case, we can use twice the resolutionof the identity to get h Φ , A Ψ i H χ ⊗H = Z B × B (cid:10) h η x , Φ i H χ , h η x , Aη y i H χ h η y , Ψ i H χ (cid:11) H d B ( x ) d B ( y )= h ˜Φ , ˜ A ˜Ψ i L ( B , H ) . Therefore, we conclude that (cid:12)(cid:12) h Φ , A Ψ i H χ ⊗H (cid:12)(cid:12) = (cid:12)(cid:12) h ˜Φ , ˜ A ˜Ψ i L ( B , H ) (cid:12)(cid:12) ≤ k ˜Φ k L ( B , H ) k ˜Ψ k L ( B , H ) k ˜ A k = k Φ k H χ ⊗H k Ψ k H χ ⊗H k ˜ A k < ∞ , and the result follows immediately. (cid:3) Let S be an elementary normal j -group. Consider the one-point phase E θ ,defined in (5.11), and given by(7.9) E θ ( qb ) = χ θ (cid:0) C q ( b − ˜ σb ) (cid:1) =: e iθ S ( qb ) . .2. A TEMPERED PAIR FROM THE ONE-POINT PHASE 125 Recall that from Lemma 6.14, we have in the coordinates (2.4) and up to a globalconstant factor: S ( a, n ⊕ m, t ) = t sinh 2 a + ω ( n, m ) cosh a . The aim of this section it to prove that the pair ( S , S ), is tempered, admissibleand tame. For this, we consider the following decomposition of the Lie algebra s (i.e . the one we used in Equation (2.10)): s = M k =0 V k where V := a , V := l ⋆ , V := l and V := R E .
As usual, we us fix { f j } dj =1 , a basis of l ⋆ to which we associate { e j } dj =1 thesymplectic-dual basis of l , defined by ω ( f i , e j ) = δ i,j . Associated to the decompo-sition v = n ⊕ m ∈ l ⋆ ⊕ l = V , we get coordinates n j := ω ( n, e j ) , m j := ω ( f j , m ) , j = 1 , . . . , d . From the expressions (2.8) of the left invariant vector fields of S , in the chart (2.4),we get the following coordinates system on S : x := e H S = 2 e − a t − (1 + e − a ) ω ( n, m ) , x j := e f j S = (1 + e − a ) m j , (7.10) x j := e e j S = (1 + e a ) n j , x := e E S = sinh 2 a . We then deduce:
Lemma . The pair ( S , S ) is tempered in the sense of Definition 1.22. More-over, the Jacobian of the map φ : S → s ⋆ , g (cid:2) s → R , X ∈ s (cid:0) e X S (cid:1) ( g ) (cid:3) , is proportional to m × ∆ − / ( d +1) S . The following Lemma is actually all that we need to prove admissibility (in thesense of Definition 1.24) of the tempered pair ( S , S ): Lemma . For every k ∈ { , , , } , there exists a tempered function m k > with ∂ x j m k = 0 for every j ≤ k and such that for every X ∈ U ( V ( k ) ) , there exists C X > with (cid:12)(cid:12) e X x k (cid:12)(cid:12) ≤ C X m k (1 + | x k | ) . Proof.
From the computations, for k ∈ N ∗ and i, j = 1 , . . . , d : e H k x = ( − k (cid:0) k +1 e − a t − k (1 + 2 k e − a ) ω ( n, m ) (cid:1) , e H k x j = ( − k (cid:0) k e − a (cid:1) m j , e f i x j = 0 , e H k x j = (cid:0) ( − k + e a (cid:1) n j , e f i x j = (1 + e a ) δ ji , e e i x j = 0 , e H k x = 2 k +1 ( cosh 2 a , k evensinh 2 a , k odd , e f i x = 0 , e e i x = 0 , e E x = 0 ,
26 7. DEFORMATION OF C ∗ -ALGEBRAS and elementary estimates, we obtain: | e Xx | ≤ C X (1 + | x | )(1 + | x | | x | ) , ∀ X ∈ U ( V ) , | e Xx j | ≤ C X (1 + | x j | ) , ∀ X ∈ U ( V (1) ) , | e Xx j | ≤ C X (1 + | x j | )(1 + | x | ) , ∀ X ∈ U ( V (2) ) , | e Xx | ≤ C X (1 + | x | ) , ∀ X ∈ U ( s ) , and the claim follows with m ( x ) = (1 + | x | | x | ), m ( x ) = 1, m ( x ) = (1 + | x | ), m ( x ) = 1. (cid:3) Repeating the arguments of the proof of Proposition 2.26, we deduce admissi-bility for the tempered pair ( S , S ). Lemma . Define X := 1 − H ∈ U ( V ) , X := 1 − d X j =1 f j ∈ U ( V ) ,X := 1 − d X j =1 e j ∈ U ( V ) , X := 1 − E ∈ U ( V ) . Then the corresponding multipliers α k := E − e X k E satisfy conditions (i) and (ii)of Definition 1.24, with ρ k = 2 and the µ k ’s are given by the m k ’s of Lemma 7.11. Lastly, we observe that tameness (see Definition 1.34) follows from Lemma 2.27and arguments very similar to those of Corollary 2.28. We then summarize all thisby stating the main result of this section:
Theorem . Let S be an elementary normal j -group and let S ∈ C ∞ ( S ) beas given in (7.9) . Then the pair ( S , S ) is tempered, admissible and tame. Remark . For B a generic normal j -group B , we could also define a one-point tempered pair, by setting(7.11) E B θ := exp { iθ S B } : B → U (1) , S B : B → R , g N X j =1 S S j ( g j ) , where S S j is the one-point phase (7.9) of each elementary factor of B in theparametrization g = g . . . g N ∈ B , relative to a Pyatetskii-Shapiro decomposi-tion. Then temperedness and admissibility will follow from arguments very similarthan those of Theorem 2.35. Remark . The one-point Schwartz space S S can ( S ) associated with the two-point pair ( S × S , S can ) as given in Definition 1.48, coincides with the one-pointSchwartz space S S ( S ) associated with the one-point par ( S , S ). Associated to a tempered, admissible and tame pair (
G, S ), we have constructedin section 1.3 a continuous linear map for any Fr´echet space E and any element m ∈ B µ ( G ) (with µ a tempered weight on G ): ^ Z G E m : B µ ( G, E ) → E , .3. EXTENSION OF THE OSCILLATORY INTEGRAL 127 which extends the ordinary integral on D ( G, E ) and that we called the oscillatoryintegral .The aim of the present section is to explain how for the tempered pair ( S , S ) ofTheorem 7.13, one can enlarge the domain of definition of the oscillatory integral.For this let E be a Fr´echet space, µ = { µ j } j ∈ N a family of tempered weights on S and ν be a fixed tempered and Y -right-invariant weight on S . Let us then considerthe following subspace of C ∞ ( S , E ): B µ,ν ( S , E ) := n F ∈ C ∞ ( S , E ) : ∀ ( j, X, Y ) ∈ N × U ( q ) × U ( Y ) , ∃ C : k e X e Y F ( qb ) k j ≤ C ν ( q ) deg( X ) µ j ( qb ) o . This space may be understood as a variant of the symbol space B µ ( S , E ), wherea specific dependence of the family of weights µ in the degree of the derivative isallowed. We endow the latter space with the following set of semi-norms:(7.12) k F k j,k ,k ,µ,ν := sup X ∈ U k ( q ) sup Y ∈ U k ( Y ) sup qb ∈ S n k e X e Y F ( qb ) k j µ j ( qb ) ν ( q ) k | X | k | Y | k o , where j, k , k ∈ N , and S k ∈ N U k ( q ), S k ∈ N U k ( Y ) are the filtrations of U ( q ) and U ( Y ) associated to the choice of PBW basis as explained in (0.4). As expected,the space B µ,ν ( S , E ) is Fr´echet for the topology induced by the semi-norms (7.12)and most the properties of Lemma 1.12 remain true. Lemma . Let ( S , E ) be as above, let µ , ρ and ˆ µ be three families of weightson S and let ν , λ and ˆ ν be three right- Y -invariant weights on S . (i) The space B µ,ν ( S , E ) is Fr´echet. (ii) The bilinear map: B µ,ν ( S ) × B ρ,λ ( S , E ) → B µ.ρ,ν.λ ( S , E ) , ( u, F ) [ g ∈ S u ( g ) F ( g ) ∈ E ] , is continuous. (iii) If there exists
C > such that µ ≤ C ˆ µ and ν ≤ C ˆ ν , then B µ,ν ( S , E ) ⊂B ˆ µ, ˆ ν ( S , E ) continuously. (iv) Assume that µ ≺ µ and ν ≺ ˆ ν . Then, the closure of D ( S , E ) in B ˆ µ, ˆ ν ( S , E ) contains B µ,ν ( S , E ) . In particular, D ( S , E ) is a dense subset of B µ,ν ( S , E ) for the induced topology of B ˆ µ, ˆ ν ( S , E ) . Proof.
The first assertion follows from the fact that a countable projec-tive limit of Fr´echet spaces is Fr´echet and that B µ,ν ( S , E ) can be realized as thecountable projective limit of the family of Banach spaces underlying the norms P ji =0 P k l =0 P k l =0 k . k i,l ,l ,µ,ν . The proof of all the other statements are identicalto their counter-parts in Lemma 1.12. (cid:3) We are now able to prove our extension result for the oscillatory integral asso-ciated to the admissible, tempered and tame pair ( S , S ): Theorem . Let µ be family of tempered weights on S , ν a Y -right-invarianttempered weight on S and m an element of B λ ( S ) for another tempered weight λ on S . Let also D ~r , ~r ∈ N , be the differential operator constructed in (1.22) . Then for We may view ν as a function on Q .
28 7. DEFORMATION OF C ∗ -ALGEBRAS all j ∈ N , there exist ~r j ∈ N , C j > and k j , l j ∈ N , such that for every element F ∈ B µ,ν ( S , E ) , we have Z S k D ~r m ( g ) F ( g ) k j d S ( g ) ≤ C j k F k j,k j ,l j ,µ,ν . Consequently, the oscillatory integral constructed in Definition 1.31 for the tameand admissible tempered pair ( S , S ) , originally defined in B µ ( S , E ) , extends as acontinuous map: ^ Z S m E : B µ,ν ( S , E ) → E . Proof.
The proof is very similar to those of Proposition 1.28, so we focus onthe differences due to the particular behavior at infinity of an element of B µ,ν ( S , E ).By Lemma 7.10, the Radon-Nikodym derivative of the left Haar measure on S with respect to the Lebesgue measure on s ⋆ , is bounded by a polynomial of order2 d + 4 in the coordinate x . For each j ∈ N , the weight µ j is also bounded bya polynomial in x , x , x , x . Now, observe that by construction of the operator D ~r in (1.22), we have for any ~r = ( r , r , r , r ) ∈ N , with K = 2 r + 2 r , K = 2 r + 2 r and with the notations given in (1.24): | D ~r F | ≤ | Ψ | | Ψ , | | Ψ , , | | Ψ , , , | (cid:12)(cid:12) e X ′ , , , F (cid:12)(cid:12) ≤ C | Ψ | | Ψ , | | Ψ , , | | Ψ , , , | µ j ν r +2 r k F k j,K ,K ,µ,ν . (7.13)This will gives the estimate we need, if we prove that the function in front of k F k j,K ,K ,µ,ν in (7.13) is integrable for a suitable choice of ~r ∈ N . We prove astronger result, namely that given ~R ∈ N , there exists ~r ∈ N such that | Ψ | | Ψ , | | Ψ , , | | Ψ , , , | ν r +2 r ≤ C (1 + | x | ) R (1 + | x | ) R (1 + | x | ) R (1 + | x | ) R . From Corollary 1.27 and Lemma 7.12, we obtain the following estimation: | Ψ | | Ψ , | | Ψ , , | | Ψ , , , |≤ C (1 + | x || x | ) r (1 + | x | ) r | x | ) r (1 + | x | ) r ( r + r + r ) (1 + | x | ) r | x | ) r . Lastly (this is the main difference with the proof of Proposition 1.28), note that ν , the tempered function on Q , can be bounded by | x | p | x | p for some integers p , p . Hence | Ψ | | Ψ , | | Ψ , , | | Ψ , , , | ν r +2 r is smaller than C (1 + | x | ) − r (1 + | x | ) − r +2 r (1 + | x | ) − r +2 r +2 p ( r + r ) × (1 + | x | ) − r +2 r ( r + r + r )+2 p ( r + r ) , and the claim follows. (cid:3) For j = 1 , . . . , N , fix m j a Y j -right-invariant tempered weight on S j (that weidentify in a natural manner as a function on Q j ), in the sense of Definition 1.17for the tempered pair ( S j , S S j ) underlying Theorem 7.13. Let also A be a C ∗ -subalgebra of B ( H ), with H a separable Hilbert space. Our aim here is to provethat for F ∈ B ( B , A ), the operator Ω θ, m ( F ), defined via a suitable quadratic form .4. A CALDER ´ON-VAILLANCOURT TYPE ESTIMATE 129 on H χ ⊗ H , is bounded . We start by proceeding formally, in order to explainour global strategy. Also, to simplify the notations, we assume first that B = S iselementary. So let Φ , Ψ ∈ H χ ⊗ H . Using twice the resolution of the identity ofProposition 7.6, we write h Φ , Ω θ, m ( F )Ψ i H χ ⊗H = Z S × S (cid:10) h η x , Φ i H χ , h η x , Ω θ, m ( F ) η y i H χ h η y , Ψ i H χ (cid:11) H d S ( x ) d S ( y ) . Next, we use the S -covariance of the pseudo-differential calculus to get h η x , Ω θ, m ( F ) η y i H χ = h η y − x , Ω θ, m ( L ⋆y − F ) η i H χ . Then, we exchange the integrals over S and Q and expand the scalar product of H χ to obtain: h η x , Ω θ, m ( F ) η i H χ = Z S × Q F ( q qb ) η x ( q ) m ( q − ) E ( qb ) η (cid:0) q s e q (cid:1) d Q ( q ) d S ( qb ) . Given F ∈ B ( S , A ), this suggests to define the function(7.14) F η : S × Q → A , ( qb, q ) F ( q qb ) η (cid:0) q s e q (cid:1) , so that with ˆ m ( qb ) := m ( q − ) and with the notations of Proposition 7.3, we willhave h η x , Ω θ, m ( F ) η i H χ = F η (cid:16) Z S E ( y ) ˆ m ( y ) F η ( ., y ) d S ( y ) (cid:17) ( x ) . Consequently, we obtain h Φ , Ω θ, m ( F )Ψ i H χ ⊗H = Z S × S (cid:10) h η x , Ψ i H χ , F η (cid:16) Z S E ( z ) ˆ m ( z ) (cid:0) L ⋆y − F (cid:1) η ( ., z ) d S ( z ) (cid:17) ( y − x ) (cid:10) η y , Ψ i H χ (cid:11) H × d S ( x ) d S ( y ) . Surprisingly, this is the right hand side of the (formal) equality above which givesrise to a well defined and bounded quadratic form on H χ ⊗ H , once the integralsign in the middle is replaced by an oscillatory one in the sense of Theorem 7.17for the tempered pair ( S , S ).Coming back to the case of a generic normal j -group B , the most importantstep is to understand the properties of the corresponding map F F η given in(7.14). Lemma . Let A be a C ∗ -algebra, B be a normal j -group with Pyatetskii-Shapiro decomposition B = ( S N ⋉ . . . ) ⋉ S and η ∈ D ( Q N × · · · × Q ) . Then themap F F η := h q N b N ∈ S N h q N − b N − ∈ S N − . . . h q b ∈ S h ( q ′ N , . . . , q ′ ) ∈ Q N × · · · × Q F ( q ′ q b . . . q ′ N − q N − b N − q ′ N q N b N ) × η (cid:0) q ′ N s e q N , . . . , q ′ s e q (cid:1) ∈ A ii . . . ii , Observe that this property holds for F ∈ S S B can ( B , A ), by Proposition 6.29 as S S B can ( B , A ) ⊂S ( B ; A ).
30 7. DEFORMATION OF C ∗ -ALGEBRAS is continuous from B ( B , A ) to B µ N ,ν N (cid:0) S N , B µ N − ,ν N − (cid:0) S N − , . . . B µ ,ν (cid:0) S , S ( Q N × · · · × Q , A ) (cid:1) . . . (cid:1)(cid:1) , where for j = N, . . . , we have settled: ν j := d Q j , µ j := (cid:8) d n j ( k j − ,l j − ; ... ; k ,l ; k,l ) S j (cid:9) ( k j − ,l j − ; ... ; k ,l ; k,l ) ∈ N j , where ( k j − , l j − ; . . . ; k , l ; k, l ) ∈ N j labels the semi-norms of the space B µ j − ,ν j − (cid:0) S j − , . . . B µ ,ν (cid:0) S , S ( Q N × · · · × Q , A ) (cid:1) . . . (cid:1) , and the exponent n j ( k j − , l j − ; . . . ; k , l ; k, l ) ∈ N is linear in its arguments. Proof.
For notational convenience, we assume that B contains only two el-ementary factors, i.e . B = S ⋉ S with S , S elementary normal j -groups. Thisis enough to understand the global mechanism and the proof for a generic normal j -group with an arbitrary number of elementary factors will then follow by induc-tion, without essential supplementary difficulties. In this simplified case, we haveto prove that the map F F η := (cid:2) q b ∈ S (cid:2) q b ∈ S (cid:2) ( q ′ , q ′ ) ∈ Q × Q F ( q ′ q b q ′ q b ) η (cid:0) q ′ s e q , q ′ s e q (cid:1) ∈ A (cid:3)(cid:3)(cid:3) , is continuous from B ( B , A ) to B µ ,ν (cid:0) S , B µ ,ν (cid:0) S , S ( Q × Q , A (cid:1)(cid:1) . (7.15)By the discussion following Remark 6.12, it is clear that one may regard S ( Q × Q , A ) as a Fr´echet space for the topology induced by the following countable setof semi-norms: k f k k,j := sup Z ∈ U k ( q ) sup Z ∈ U k ( q ) sup ( q ,q ) ∈ Q × Q n d Q ( q ) j d Q ( q ) j k Z q Z q f ( q , q ) k| Z | k | Z | k o , i.e. we may use right-invariant vector fields instead of left-invariant one since theyare related by tempered functions with tempered inverses. Note then that the nat-ural Fr´echet topology of the space (7.15) is associated with the following countablefamily of semi-norms (indexed by ( k , l , k , l , k, j ) ∈ N ):Φ sup X ∈ U k ( q ) sup X ′ ∈ U l ( Y ) sup q b ∈ S sup Y ∈ U k ( q ) sup Y ′ ∈ U l ( Y ) sup q b ∈ S sup Z ∈ U k ( q ⊕ q ) sup ( q ,q ) ∈ Q × Q d Q ( q ′ ) j d Q ( q ′ ) j k Z ( q ′ ,q ′ ) e Y q b e Y ′ q b e X q b e X ′ q b Φ( q b ; q b ; q ′ , q ′ ) k d S ( q b ) n ( k ,l ,k,j ) d Q ( q ) k d S ( q b ) n ( k,j ) d Q ( q ) k | X | k | X ′ | l | Y | k | Y ′ | l | Z | k . To simplify the notations, we denote the latter semi-norm by k . k k ,l ,k ,l ,k,j . Then,for ( X, X ′ , Y, Y ′ , Z , Z ) ∈ U ( q ) × U ( Y ) × U ( q ) × U ( Y ) × U ( q ) × U ( q ) , we get within Sweedler’s notation: Z q ′ Z q ′ e Y q b e Y ′ q b e X q b e X ′ q b F η ( q b ; q b ; q ′ , q ′ ) = X ( X ) X ( X ′ ) X ( Y ) X ( X ′ ) X ( Z ) X ( Z ) (cid:16) Z q ′ Z q ′ e Y (1) q b e Y ′ (1) q b e X (1) q b e X ′ (1) q b F ( q ′ q b q ′ q b ) (cid:17) × (cid:16) Z q ′ Z q ′ e Y (2) q e X (2) q η (cid:0) q ′ s e ( q ) , q ′ s e ( q ) (cid:1)(cid:17) . .4. A CALDER ´ON-VAILLANCOURT TYPE ESTIMATE 131 From the same reasoning as those in the proof of Lemma 1.8 (v), we deduce that k F η k k ,k ,k,j ≤ C sup d Q ( q ′ ) j d Q ( q ′ ) j d S ( q b ) n ( k ,l ,k,j ) d Q ( q ) k d S ( q b ) n ( k,j ) d Q ( q ) k × sup (cid:13)(cid:13) Z q ′ Z q ′ e Y q b e Y ′ q b e X q b e X ′ q b F ( q ′ q b q ′ q b ) (cid:13)(cid:13) | X | k | X ′ | l | Y | k | Y ′ | l | Z | k | Z | k × sup (cid:12)(cid:12) Z q ′ Z q ′ e Y q e X q η (cid:0) q ′ s e ( q ) , q ′ s e ( q ) (cid:1)(cid:12)(cid:12) | X | k | Y | k | Z | k | Z | k , (7.16)where the first supremum is over:( q b , q b , q ′ , q ′ ) ∈ S × S × Q × Q , the second over:( X, X ′ , Y, Y ′ , Z , Z ) ∈ U k ( q ) × U l ( Y ) × U k ( q ) × U l ( Y ) × U k ( q ) × U k ( q ) , and the third over:( X, Y, Z , Z ) ∈ U k ( q ) × U k ( q ) × U k ( q ) × U k ( q ) . Next, we observe: Z q ′ Z q ′ e Y q b e Y ′ q b e X q b e X ′ q b F ( q ′ q b q ′ q b ) = (cid:0) ^ Ad ( q ′ q b q ′ q b ) − ( Z ) ^ Ad ( q ′ q b ) − ( Z ) ^ Ad ( q ′ q b ) − ( Y Y ′ ) e X e X ′ F (cid:1) ( q ′ q b q ′ q b ) . This, together with Lemma 1.14, entails that the F -dependent supremum in (7.16)is, up to a constant, bounded by: k F k k + k + l + k + l d S ⋉S ( q ′ q b q ′ q b ) k d S ⋉S ( q ′ q b ) k + k + l , which by sub-multiplicativity of the modular weight, is bounded by: k F k k + k + l + k + l d S ⋉S ( q ′ q b ) k + k + l d S ⋉S ( q ′ q b ) k . Now, by Theorem 2.35, the pair (cid:0) ( S ⋉ S ) , S S can ⊕ ⊕ S S can (cid:1) is tempered(and admissible and tame), so that by Lemma 1.21 the modular weight d ( S ⋉S ) is tempered. Then, using the last statement of Lemma 1.5, together with themethods of Lemmas 2.33 and 2.34, we see that d S ⋉S is tempered in (any) adaptedcoordinates (see Definition 2.32) for B = S ⋉ S . This clearly implies that therestriction d S ⋉S | S j , j = 1 ,
2, is also tempered in the adapted coordinates for S j .In view of the expressions (7.10), we see that the adapted tempered coordinatesand the coordinates associated to the one-point pair ( S j , S S j ) (which is temperedby Theorem 7.13) are related to one another through a tempered diffeomorphism.Hence, we deduce that d S ⋉S | S j is tempered in the sense of the one-point phasefunction too. Last, using the explicit expression of the tempered weight d S j given inLemma 2.27, we deduce that there exist m j ∈ N and C j >
0, such that d S ⋉S | S j ≤ C j d m j S j , j = 1 ,
2. Hence, the F -dependent supremum in (7.16) is, up to a constant,bounded by k F k k + k + l + k + l d S ( q ′ q b ) m (2 k + k + l ) d S ( q ′ q b ) m k . For η -dependent term in (7.16), we first note: Z q ′ Z q ′ e Y q e X q η (cid:0) q ′ s e ( q ) , q ′ s e ( q ) (cid:1) = e Y q e X q (cid:0) Z Z η (cid:1)(cid:0) q ′ s e ( q ) , q ′ s e ( q ) (cid:1) ,
32 7. DEFORMATION OF C ∗ -ALGEBRAS so that up to a redefinition of η , we can ignore the right-invariant vector fields.Next, we observe that with q = ( a, n ), q ′ = ( a ′ , n ′ ) in the coordinates (6.10), wehave: q ′ s e ( q ) = (cid:0) a + a ′ , e − a n ′ + 2 n cosh a (cid:1) . With H the generator of a and { f j } dj =1 a basis of l ⋆ , the associated left-invariantvector fields on Q read: e H = ∂ a − d X j =1 n j ∂ n j , e f j = ∂ n j . Choosing η = η ⊗ η with η j ∈ D ( Q j ), it is enough to treat each variable separately.So just assume that η ∈ D ( Q ). Now, for N = ( N , . . . , N d ) ∈ N d with | N | = k , wehave with e f N := e f N . . . e f N d d , ∂ Nn := ∂ N n . . . ∂ N d n d and setting q := q ′ s e ( q ) ∈ Q : e f Nq η ( q ) = 2 k cosh k a (cid:0) ∂ Nn η (cid:1) ( q ) . Since cosh a ≤ a ′ / a /
2, the latter and Remark 6.12 entail that (cid:12)(cid:12) e f Nq η ( q ) (cid:12)(cid:12) ≤ C cosh( a ′ / k cosh( a / k (cid:12)(cid:12) ∂ Nn η (cid:12)(cid:12) ( q ) ≤ C d Q ( q ′ ) k cosh( a / k (cid:12)(cid:12) ∂ Nn η (cid:12)(cid:12) ( q ) . On the other hand, we have e H q η ( q ) = 2 (cid:0) ∂ a η (cid:1) ( q ) − d X j =1 ( n j e − a + n ′ j e − a ) (cid:0) ∂ n j η (cid:1) ( q ) . Since w j := n j e − a + n ′ j e − a is an eigenvector of e H q with eigenvalue −
2, we deducethat for k ∈ N , e H kq η ( q ) is a linear combinations of the ordinary derivatives of η ,with coefficients given in the ring C [ w j ] of order at most k . Moreover, the roughestimate: | w | = | ( w , . . . , w d ) | ≤ a cosh a ′ ( | n | + | n ′ | ) , gives by Remark 6.12: (cid:12)(cid:12) e H kq η ( q ) (cid:12)(cid:12) ≤ C d Q ( q ′ ) k cosh k a | n | k (cid:12)(cid:12) P ( ∂ a , ∂ n j ) η (cid:12)(cid:12) ( q ) , for a suitable polynomial P . This implies that the η -dependent term in (7.16) is,up to a constant, bounded by: d Q ( q ′ ) k d Q ( q ′ ) k | ˜ η | (cid:0) q ′ s e ( q ) , q ′ s e ( q ) (cid:1) , where ˜ η belongs to D ( Q × Q ) and is obtained from η by multiplication by cosh a, | n | and by differentiation along all its variables. Finally, we deduce (with m , m ∈ N fixed) that k F η k k ,l ,k ,l ,k,j ≤ C k F k k + k + l + k + l sup ( q b ,q b ,q ′ ,q ′ ) ∈ S × S × Q × Q d Q ( q ′ ) j + k d Q ( q ′ ) j + k d S ( q ′ q b ) m (2 k + k + l ) d S ( q ′ q b ) m k | ˜ η | (cid:0) q ′ s e ( q ) , q ′ s e ( q ) (cid:1) d S ( q b ) n ( k ,l ,k,j ) d Q ( q ) k d S ( q b ) n ( k,j ) d Q ( q ) k . Observe then that by Lemma 2.27 and Remark 6.12, we have d S | Q ≤ C d Q . Then, bythe sub-multiplicativity and the invariance under the inversion map of the modular .4. A CALDER ´ON-VAILLANCOURT TYPE ESTIMATE 133 weights, we deduce that the fraction above is smaller than (a constant times): d Q (cid:0) s e ( q ) (cid:1) j + k +2 m (2 k + k + l ) d Q (cid:0) s e ( q ) (cid:1) j + k +2 m k d S ( q b ) m (2 k + k + l ) d S ( q b ) n ( k ,l ,k,j ) d Q ( q ) k d S ( q b ) n ( k,j ) d Q ( q ) k × d S ( q b ) m k d Q (cid:0) q ′ s e ( q ) (cid:1) j + k +2 m (2 k + k + l ) d Q (cid:0) q ′ s e ( q ) (cid:1) j + k +2 m k × | ˜ η | (cid:0) q ′ s e ( q ) , q ′ s e ( q ) (cid:1) . Because ˜ η is compactly supported, the expression in the last line above is smallerthan a constant. Also, since s e ( a, n ) = (2 a, n cosh a ), we deduce, by Remark 6.12again, that d Q ◦ s e ≤ C d Q . Thus, the expression above is bounded by (a constanttimes): d Q ( q ) j +4 m (2 k + k + l ) d Q ( q ) j +4 m k d S ( q b ) m (2 k + k + l ) d S ( q b ) m k d S ( q b ) n ( k ,l ,k,j ) d S ( q b ) n ( k,j ) , and one concludes using Lemma 2.27 and Remark 6.12, which show that for all q ∈ Q , b ∈ Y , we have d Q ( q ) ≤ d S ( qb ) and, by suitably choosing n ( k, j ) and n ( k , l , k, j ). (cid:3) Remark . Lemma (7.18) admits a straightforward generalization for sym-bols valued in a Fr´echet algebra. Namely, if E is a Fr´echet algebra and µ is a familyof tempered weights on B , then the map F F η is continuous from B µ ( B , E ) to B µ N ,ν N (cid:0) S N , B µ N − ,ν N − (cid:0) S N − , . . . B µ ,ν (cid:0) S , S ( Q N × · · · × Q , A ) (cid:1) . . . (cid:1)(cid:1) , where the µ j ’s now depend also of the restriction of µ to S j (and the ν j ’s areunchanged).We are now ready to prove a non-Abelian (curved) and C ∗ -valued version ofthe Calder´on-Vaillancourt estimate, the main result of this chapter. Theorem . Let B be a normal j -group, A a C ∗ -algebra faithfully repre-sented on a separable Hilbert space H , F ∈ B ( B , A ) , η ∈ D ( Q N × · · · × Q ) and m ∈ Θ ( B ) . Define for x, y ∈ B , the element of A given by h η x , Ω θ, m ( F ) η y i H χ := F η (cid:16) ^ Z S E S θ ˆ m . . . (cid:16) ^ Z S N E S N θ ˆ m N h g N ∈ S N . . . h g ∈ S (cid:0) L ⋆y − F (cid:1) η ( g N , . . . , g ; . ) i . . . i(cid:17) . . . (cid:17) ( y − x ) , (7.17) where ˆ m ( qb ) = m ( q − ) and where E S j θ is the one-point phase of S j as defined in (7.9) . Then we have: sup x ∈ B Z B (cid:13)(cid:13) h η x , Ω θ, m ( F ) η y i H χ (cid:13)(cid:13) d B ( y ) < ∞ , sup y ∈ B Z B (cid:13)(cid:13) h η x , Ω θ, m ( F ) η y i H χ (cid:13)(cid:13) d B ( x ) < ∞ . Consequently (see Proposition 7.9), the operator Ω θ, m ( F ) on H χ ⊗ H defined bymeans of the quadratic form Ψ , Φ ∈ H χ ⊗ H 7→ Z B × B (cid:10) h η x , Ψ i H χ , h η x , Ω θ, m ( F ) η y i H χ h η y , Ψ i H χ (cid:11) H d B ( x ) d B ( y ) , is bounded. Moreover, there exists k ∈ N (depending only on dim B and on theorder of the polynomial in d Q N ⊗ · · · ⊗ d Q that majorizes | m | ) and C > , such
34 7. DEFORMATION OF C ∗ -ALGEBRAS that for all F ∈ B ( B , A ) we have k Ω θ, m ( F ) k ≤ C k F k k, ∞ = C sup X ∈U k ( b ) sup x ∈ B (cid:13)(cid:13) e X F ( x ) (cid:13)(cid:13) . Proof.
To simplify the notation for the matrix element given in (7.17), wewrite h η x , Ω θ, m ( F ) η y i H χ = F η (cid:16) ^ Z B E B θ ˆ m (cid:2) z (cid:0) L ⋆y − F (cid:1) η ( ., z ) (cid:3)(cid:17) ( y − x ) , where E B θ is given in (7.11). Observe that this notation is coherent with our Fubinitype Theorem 1.39. Thus,sup y ∈ B Z B (cid:13)(cid:13) h η x , Ω θ, m ( F ) η y i H χ (cid:13)(cid:13) d B ( x )= sup y ∈ B Z B (cid:13)(cid:13) F η (cid:16) ^ Z B E B θ ˆ m (cid:2) z (cid:0) L ⋆y − F (cid:1) η ( ., z ) (cid:3)(cid:17) ( y − x ) (cid:13)(cid:13) d B ( x )= sup y ∈ B Z B (cid:13)(cid:13) F η (cid:16) ^ Z B E B θ ˆ m (cid:2) z (cid:0) L ⋆y − F (cid:1) η ( ., z ) (cid:3)(cid:17) ( x ) (cid:13)(cid:13) d B ( x ) . The fact that this expression is finite follows then by combining Propositions 7.3and 7.5 with Theorem 7.17 and Lemma 7.18 and the fact that L ⋆y − maps B ( B , A )to itself isometrically. The second case is similar since h η x , Ω θ, m ( F ) η y i ∗H χ = h η y , Ω θ,σ ⋆ m ( F ∗ ) η x i H χ . The final estimation we give is a consequence of Proposition 7.9 together with theestimates underlying Lemma 7.5, Theorem 7.17 and Lemma 7.18. (cid:3)
Remark . In view of Remark 7.19, on may wonder what happens in The-orem 7.20 when one choses a symbol in B µ ( B , A ) instead of a symbol in B ( B , A ).So let F ∈ B µ ( B , A ), with µ a tempered weight. Then, from the same argumentthan those of Theorem 7.20 (using by Remark 7.19, instead of Lemma 7.18), onededuces that Z B (cid:13)(cid:13) h η x , Ω θ, m ( F ) η y i H χ (cid:13)(cid:13) d B ( x ) < ∞ . (7.18)However, as the left regular action is no longer isometric on B µ ( B , A ) (see forinstance the second item of Lemma 1.12), there is no reason to expect that thesupremum over y ∈ B of the expression given in (7.18) to be finite. Accordingly(and as expected), there is no chance for the operator Ω θ, m ( F ) to be bounded when F belongs to B µ ( B , A ) with unbounded µ . C ∗ -norm Now, we assume that our C ∗ -algebra A is equipped with a strongly continuousand isometric action α of a normal j -group B . We stress that the results of thischapter cannot hold true in the more general context of tempered actions. Thisis the main difference between the deformation theory at the level of Fr´echet and C ∗ -algebras. Given an element a ∈ A , we construct as usual the A -valued function α ( a ) on B : α ( a ) := [ g ∈ B α g ( a ) ∈ A ] . .5. THE DEFORMED C ∗ -NORM 135 Thus, from Theorem 4.8, we can deform the Fr´echet algebra structure on the setof smooth vectors A ∞ by means of the deformed product a ⋆ αθ, m b := (cid:0) α ( a ) ⋆ θ, m α ( b ) (cid:1) ( e ) , a, b ∈ A ∞ . We have seen in (6.15) how to modify the original involution at the level of B ( B , A ).At the level of the Fr´echet algebra A ∞ , an obvious observation leads to: Lemma . Let B be a normal j -group. For m ∈ Θ ( B ) , the following definesa continuous involution of the Fr´echet algebra ( A ∞ , ⋆ αθ, m ) : ∗ θ, m : A ∞ → A ∞ , a m N s ⋆e m N (cid:16) arcsinh( iθ E αN ) (cid:17) . . . m s ⋆e m (cid:16) arcsinh( iθ E α ) (cid:17) a ∗ , where E N , . . . , E are the central elements of the Heisenberg Lie algebras attachedto each elementary factors of B . Remark . Note that when s ⋆e m j = m j , j = 1 , . . . , N , there is no modifi-cation of the involution.The construction of a pre- C ∗ -structure on ( A ∞ , ⋆ αθ, m ) follows then from Theo-rem 7.20 and from the following immediate result (compare with Lemma 4.5): Lemma . Let ( A, α, B ) be a C ∗ -algebra endowed with a strongly continuousand isometric action of a normal j -group. Then, we have an isometric equivariantembedding α : A ∞ → B ( B , A ) . Proof.
The equivariance property of α is obvious and implies (with the factthat α is an isometric action of B on A ) that for any k ∈ N : k α ( a ) k k, ∞ = sup g ∈ B sup X ∈ U k ( b ) k e X g α g ( a ) k| X | k = sup g ∈ B sup X ∈ U k ( b ) k α g ( X α a ) k| X | k = sup X ∈ U k ( b ) k X α a k| X | k = k a k k , and the proof follows. (cid:3) Recall that H is any separable Hilbert space carrying a faithful representationof our C ∗ -algebra A , which therefore, is identified with a C ∗ -subalgebra of B ( H ).Then by the previous lemma and Theorem 7.20, we deduce that the map a ∈ A ∞
7→ k Ω θ, m (cid:0) α ( a ) (cid:1) k , takes finite values. It is also important to observe that the norm above is byconstruction the operator norm on H χ ⊗ H , that is the spatial C ∗ -norm on A ⊗B ( H χ ). For future use, we also recall that the spatial C ∗ -norm is the minimal C ∗ -cross-norm. (We invite the reader unfamiliar with tensor products of C ∗ -algebrasto consult, for example, Appendix B of [ ].) More precisely, combining the Lemma7.24 with Theorem 7.20, we deduce the following inequality: Corollary . Let ( A, α, B ) be a C ∗ -algebra endowed with a strongly con-tinuous action of a normal j -group and m ∈ Θ ( B ) . Then, there exists k ∈ N and C > such that for any a ∈ A ∞ , we have: k Ω θ, m (cid:0) α ( a ) (cid:1) k ≤ C k a k k := C sup X ∈ U k ( b ) n k X α a k| X | k o .
36 7. DEFORMATION OF C ∗ -ALGEBRAS Proposition . Let m ∈ Θ ( B ) . Then, the following defines a C ∗ -norm onthe involutive deformed Fr´echet algebra ( A ∞ , ⋆ αθ, m , ∗ θ, m ) : a ∈ A ∞
7→ k a k θ, m := (cid:13)(cid:13) Ω θ, m (cid:0) α ( a ) (cid:1)(cid:13)(cid:13) , where the operator Ω θ, m (cid:0) α ( a ) (cid:1) is defined in Theorem 7.20. Accordingly, we let A θ, m be the C ∗ -completion of A ∞ that we abusively call the C ∗ -deformation of A . Proof.
By construction we have for all a, b ∈ A ∞ Ω θ, m (cid:0) α ( a ∗ θ, m ⋆ αθ, m b ) (cid:1) = Ω θ, m (cid:0) α ( a ) (cid:1) ∗ Ω θ, m (cid:0) α ( b ) (cid:1) , and the claim follows immediately. (cid:3) Remark . We already know that at the level of the deformed pre- C ∗ -algebra ( A ∞ , ⋆ αθ, m ), the action of the group B is no longer by automorphism. Butat the level of the deformed C ∗ -algebra there is no action of B at all.In a way very analogous to Proposition 4.20, we can show that the C ∗ -deforma-tion associated with a normal j -group coincides with the iterated C ∗ -deformationsof each of its elementary normal subgroups. To see this, fix B be a normal j -groupwith Pyatetskii-Shapiro decomposition B = B ′ ⋉S and A a C ∗ -algebra endowed witha strongly continuous and isometric action α of B . Of course, α S , the restriction of α to the subgroup S , is strongly continuous on A . Let us fix also ˜ m = m ′ ⊗ m ∈ Θ ( B ),with m ′ ∈ Θ ( B ′ ) and m ∈ Θ ( S ). Then, we can perform the C ∗ -deformation of A by means of the action of S . We call this deformed C ∗ -algebra A S θ, m . Then B ′ actsstrongly continuously by ∗ -homomorphisms on A S θ, m . Indeed, it has been shown inthe proof of Proposition 4.20 that the subspace of smooth vectors for B coincideswith the subspace of smooth vectors for B ′ within the subspace of smooth vectorsfor S . In turns, A ∞ , the set of smooth vectors for B on A , is dense in A S θ, m . Asthe action of B ′ is (obviously) strongly continuous and by ∗ -homomorphisms (asshown in the proof of Proposition 4.20 too) on A ∞ , a density argument yields theresult. Thus, we can perform the C ∗ -deformation of A S θ, m by means of the actionof B ′ . We call this deformed C ∗ -algebra ( A S θ, m ) B ′ θ, m ′ . But we could also perform the C ∗ -deformation of A by means of the action of B directly. We call this deformed C ∗ -algebra A B θ, ˜ m . Now, the precise result of Proposition 4.20, is that at the level ofthe (common) dense subspace A ∞ , both constructions coincide. Thus it suffices toshow that the C ∗ -norms of ( A S θ, m ) B ′ θ, m ′ and A B θ, ˜ m coincide on A ∞ . But this easilyfollows from our construction. Indeed, the C ∗ -norm of ( A S θ, m ) B ′ θ, m ′ at the level of A ∞ , is by definition the map a (cid:13)(cid:13) Ω B ′ θ, m ′ (cid:0)(cid:2) z ′ ∈ B ′ Ω S θ, m (cid:0) [ z ∈ S α zz ′ ( a )] (cid:1)(cid:3)(cid:1)(cid:13)(cid:13) . But by the construction of section 6.3, we precisely haveΩ B ′ θ, m ′ (cid:0)(cid:2) z ′ ∈ B ′ Ω S θ, m (cid:0) [ z ∈ S α zz ′ ( a )] (cid:1)(cid:3)(cid:1) = Ω B θ, ˜ m (cid:0) α ( a ) (cid:1) . Thus, we have proved the following:
Proposition . Let B be a normal j -group with Pyatetskii-Shapiro decom-position B = B ′ ⋉ S , where B ′ is a normal j -group and S is an elementary normal j -group. Let A be a C ∗ -algebra endowed with a strongly continuous isometric action α of B . Within the notations displayed above, we have: A B θ, ˜ m = ( A S θ, m ) B ′ θ, m ′ . .5. THE DEFORMED C ∗ -NORM 137 In the remaining part of this section we prove that the deformed C ∗ -normconstructed above coincide with the C ∗ -norm of bounded and adjointable operatorson a C ∗ -module. This will make clearer the analogies with the construction ofRieffel in [ ] for the Abelian case and it also explains the choice of the spatialtensor product in Theorem 7.20. Definition . Let m ∈ Θ ( B ). Then, for f , f ∈ S S can ( B , A ), we define the A -valued pairing: h f , f i θ, m := Z B (cid:10) η x , Ω θ, m ( f ∗ θ, m ⋆ θ, m f ) η x (cid:11) d B ( x ) , (7.19)where where { η x } x ∈ B ⊂ H χ is the family of coherent states given in Definition 7.2and the involution ∗ θ, m on S S can ( B , A ) is defined by: ∗ θ, m : f m N s ⋆e m N (cid:16) arcsinh( iθ e E N ) (cid:17) ◦ · · · ◦ m s ⋆e m (cid:16) arcsinh( iθ e E ) (cid:17) f ∗ . In the last formula, E N , . . . , E denote the central elements in each Heisenberg Liealgebra attached to each elementary components in B and f ∗ := [ x ∈ B f ( x ) ∗ ] ∈S S can ( B , A ). Proposition . Endowed with the pairing (7.19) and action S S can ( B , A ) × A ∞ → S S can ( B , A ) , ( f, a ) (cid:2) g ∈ B f ( g ) a (cid:3) , the space S S can ( B , A ) becomes a (right) pre- C ∗ -module for the C ∗ -algebra A . Proof.
We need first to show that the pairing (7.19) is well defined. Also, tolighten a little bit the notations, we assume that the normal j -group B contains onlytwo elementary factors, i.e . B = S ⋉ S with S , S elementary normal j -groups.(The proof for a generic normal j -group with an arbitrary number of elementaryfactors has no essential supplementary difficulties.) Take first an element F ∈B µ ( B , E ), where E is any Fr´echet space and µ any family of tempered weights. Inthis situation (which is slightly more general than the situation of Theorem 7.20),by analogy with equation (7.17), it is natural to define: h η, Ω θ, m ( F ) η i H χ := F η (cid:16) ^ Z S E S θ ˆ m (cid:16) ^ Z S E S θ ˆ m h g ∈ S h g ∈ S F η ( g , g ; . ) ii(cid:17)(cid:17) ( e ) . By a slight adaptation of Lemma 7.18 (see Remark 7.19), we deduce that F η ∈ B µ ,ν (cid:0) S , B µ ,ν (cid:0) S , S ( Q × Q , E (cid:1)(cid:1) , for suitable tempered weights. Now, Theorem 7.17 and Proposition 7.3 entails that h η, Ω θ, m ( F ) η i H χ ∈ E . This observation being made, we further remark that for ev-ery f ∈ S S can ( B , A ), the element ˙ f ∈ C ∞ ( B , S S can ( B , A )) defined by ˙ f ( x ) := [ y f ( xy )] actually lives in B µ ( B , S S can ( B , A )) for a suitable family of tempered weights µ . Applying the preceding reasoning for the Fr´echet space E = S S can ( B , A ), we de-duce that the element h η , Ω θ, m ( ˙ f ) η i lives in S S can ( B , A ). Using the B -equivarianceof the quantization map Ω θ, m , we then notice that the value at x ∈ B of the aboveelement h η , Ω θ, m ( ˙ f ) η i equals h η x , Ω θ, m ( f ) η x i . Which we deduce from that thematrix coefficient [ x
7→ h η x , Ω θ, m ( f ) η x i ] belongs to S S can ( B , A ). Hence, Z B (cid:13)(cid:13) h η x , Ω θ, m ( f ) η x i (cid:13)(cid:13) d B ( x ) < ∞ .
38 7. DEFORMATION OF C ∗ -ALGEBRAS Thus, we conclude from the stability of S S can ( B , A ) under ⋆ θ, m (see Proposition3.10), that the pairing h ., . i θ, m is well defined.Testing this pairing on the dense subset span { aϕ , a ∈ A , ϕ ∈ S S can ( B ) } of S S can ( B , A ), we see that hS S can ( B , A ) , S S can ( B , A ) i θ, m = A.A , which is dense in A . Next, we observe that the pairing can be rewritten as h f , f i θ, m = Z B (cid:10) η x , Ω θ, m ( f ) ∗ Ω θ, m ( f ) η x (cid:11) d B ( x ) . This shows that h f , f i ∗ θ, m = h f , f i θ, m and proves positivity and non-degeneracy.Last, it is clear that h f , f i θ, m a = h f , f a i θ, m for all a ∈ A and all S S can ( B , A ). (cid:3) Remark . It can be shown that the pairing can be rewritten as: h f , f i θ, m = Z B f ∗ θ, m ⋆ θ, m f ( g ) d B ( g ) = Tr (cid:0) Ω θ, m ( f ) ∗ Ω θ, m ( f ) (cid:1) . However, this is by far less convenient expressions, as shown in the proof of Theorem7.33 below.
Definition . Let m ∈ Θ ( B ). For F ∈ B ( B , A ), let L θ, m ( F ) be the opera-tor on S S can ( B , A ) given by L θ, m ( F ) f = F ⋆ θ, m f . By Proposition 3.10, the operator L θ, m ( F ), F ∈ B ( B , A ), acts continuously on S S can ( B , A ). Moreover, L θ, m ( F ) is adjointable, with adjoint given by L θ, m ( F ∗ θ, m ).Indeed, for all f , f ∈ S S can ( B , A ) and F ∈ B ( B , A ), we have h f , L θ, m ( F ) f i θ, m = Z B (cid:10) η x , Ω θ, m ( f ∗ θ, m ⋆ θ, m F ⋆ θ, m f ) η x (cid:11) d B ( x )= Z B (cid:10) η x , Ω θ, m (cid:0) ( F ∗ θ, m ⋆ θ, m f ) ∗ θ, m ⋆ θ, m f (cid:1) η x (cid:11) d B ( x )= h L θ, m ( F ∗ θ, m ) f , f i θ, m . Note also that the operators L θ, m ( F ) all commute with the right-action of A . Butwe have more, since in fact L θ, m ( F ), for F ∈ B ( B , A ), belongs to the C ∗ -algebraof A -linear adjointable endomorphisms of the pre- C ∗ -module S S can ( B , A ). Indeed,from the operator inequality on B ( H χ ) ⊗ A Ω θ, m ( f ∗ θ, m ⋆ θ, m F ∗ θ, m ⋆ θ, m F ⋆ θ, m f ) = Ω θ, m ( f ) ∗ (cid:12)(cid:12) Ω θ, m ( F ) (cid:12)(cid:12) Ω θ, m ( f ) ≤ k Ω θ, m ( F ) k Ω θ, m ( f ) ∗ Ω θ, m ( f )= k Ω θ, m ( F ) k Ω θ, m ( f ∗ θ, m ⋆ θ, m f ) , we deduce for F ∈ B ( B , A ) and f ∈ S S can ( B , A ), the operator inequality on A : h L θ, m ( F ) f, L θ, m ( F ) f i θ, m ≤ k Ω θ, m ( F ) k h f, f i θ, m . Hence we get(7.20) k L θ, m ( F ) k ≤ k Ω θ, m ( F ) k , .6. FUNCTORIAL PROPERTIES OF THE DEFORMATION 139 where the norm on the left hand side denotes the norm of the C ∗ -algebra of A -linearadjointable endomorphisms of the pre- C ∗ -module S S can ( B , A ). Now, observe thedense embedding of the algebraic tensor product B ( B ) ⊗ alg A → B ( B , A ) , X i φ i ⊗ a i (cid:2) g ∈ B X i φ i ( g ) a i ∈ A (cid:3) . Via this embedding, the norm on the right hand side of (7.20) is by constructionthe restriction to B ( B ) ⊗ alg A of the minimal (spatial) C ∗ -norm on B ⊗ alg A , where B is the C ∗ -completion of { Ω θ, m ( F ) , F ∈ B ( B ) } in B ( H χ ). Hence, we deduce that k L θ, m ( F ) k ≥ k Ω θ, m ( F ) k , ∀ F ∈ B ( B ) ⊗ alg A , which by density implies that k L θ, m ( F ) k ≥ k Ω θ, m ( F ) k , ∀ F ∈ B ( B , A ) . Thus we have proved the following:
Theorem . Let B be a normal j -group, A a C ∗ -algebra and m ∈ Θ ( B ) .Then k L θ, m ( F ) k = k Ω θ, m ( F ) k , ∀ F ∈ B ( B , A ) , where the norm on the left hand side is the one of the C ∗ -algebra of A -linear ad-jointable endomorphisms of the pre- C ∗ -module S S can ( B , A ) and the norm on theright hand side is the spatial C ∗ -norm of B ( H χ ) ⊗ A . Back to the case where A carries a strongly continuous isometric action α , wededuce: Proposition . Let ( A, α ) be a C ∗ -algebra endowed with a strongly con-tinuous and isometric action of a normal j -group B and let m ∈ Θ ( B ) . Then, the C ∗ -norm on the involutive Fr´echet algebra ( A ∞ , ⋆ αθ, m , ∗ θ, m ) given by a ∈ A ∞ (cid:13)(cid:13) L θ, m (cid:0) α ( a ) (cid:1)(cid:13)(cid:13) , coincides with the deformed norm k . k θ, m of Proposition 7.26. In this section, we collect the main functorial properties of the deformation. Westill consider a C ∗ -algebra A , endowed with a strongly continuous and isometricaction α of a normal j -group B . Given an element m ∈ Θ ( B ), we form A θ, m , the C ∗ -deformation of A . We let B θ, m ( B , A ) and S θ, m ( B , A ) be the C ∗ -completion ofthe pre- C ∗ -algebras: (cid:0) B ( B , A ) , ⋆ θ, m , ∗ θ, m (cid:1) and (cid:0) S S can ( B , A ) , ⋆ θ, m , ∗ θ, m (cid:1) , for the (deformed) C ∗ -norm F
7→ k Ω θ, m ( F ) k . Firstly, we observe from Proposition 6.29, the following isomorphism:
Lemma . Let A be a C ∗ -algebra and m ∈ Θ ( B ) . Then we have: S θ, m ( B , A ) ≃ K ( H χ ) ⊗ A .
40 7. DEFORMATION OF C ∗ -ALGEBRAS Now, we come to the question of bounded approximate units for the deformed C ∗ -algebra A θ, m . Since A possesses a bounded approximate unit (as any C ∗ -algebradoes), Proposition 4.19 shows that the pre- C ∗ -algebra ( A ∞ , ⋆ αθ, m , ∗ θ, m ) possesses abounded approximate unit as well. Thus, we deduce from Corollary 7.25: Proposition . Let ( A, α ) be a C ∗ -algebra endowed with a strongly contin-uous action of a normal j -group B and m ∈ Θ ( B ) . Then A θ, m possesses a boundedapproximate unit { e λ } λ ∈ Λ consisting of elements of A ∞ . Next, we observe that the two-sided ideal ( S S can ( B , A ) , ⋆ θ, m , ∗ θ, m ) is essentialin ( B ( B , A ) , ⋆ θ, m , ∗ θ, m ): Proposition . Let A be a C ∗ -algebra and m ∈ Θ ( B ) . Then, the ideal S S can ( B , A ) is essential in the pre- C ∗ -algebra B ( B , A ) , that is to say we have for all F ∈ B ( B , A ) : k Ω θ, m ( F ) k = sup (cid:8) k Ω θ, m ( F ⋆ θ, m f ) k : f ∈ S S can ( B , A ) , k Ω θ, m ( f ) k ≤ (cid:9) . Proof.
This is verbatim the arguments of [ , Proposition 4.11], combinedwith the equality k Ω θ, m ( F ) k = k L θ, m ( F ) k of Proposition 7.34, for all F ∈ B ( B , A )(thus for f ∈ S S can ( B , A ) too) and with the existence of bounded approximate unitsof the pre- C ∗ -algebra ( S S can ( B , A ) , ⋆ θ, m ) as shown in Proposition 4.19. (cid:3) The proof of the next two results is word for word the one of the correspondingresults in the flat situation, given in [ , Proposition 4.12 and Proposition 4.15]. Proposition . Let A be a C ∗ -algebra, I an essential ideal of A and m ∈ Θ ( B ) . Then the C ∗ -norm on (cid:0) B ( B , A ) , ⋆ θ, m (cid:1) given in Theorem 7.20 is the sameas the C ∗ -norm of Proposition 7.34 for the restriction of the action of B ( B , A ) on S S can ( B , I ) . Proposition . Let A be a C ∗ -algebra and m ∈ Θ ( B ) . The C ∗ -algebra B θ, m ( B , A ) is isomorphic to the C ∗ -deformation of the algebra of A -valued rightuniformly continuous and bounded functions on B , C ru ( B , A ) , for the right regularaction of B . The following two results treat the question of morphisms and ideals. Theycan be proved exactly as [ , Theorem 5.7, Proposition 5.8 and Proposition 5.9],by using our Propositions 4.15 and 4.17. Proposition . Fix m ∈ Θ ( B ) and let ( A, α ) and ( B, β ) be two C ∗ -algebrasendowed with strongly continuous actions of B . Then, if T : A → B is a continuoushomomorphism which intertwines the actions α and β , its restriction T ∞ : A ∞ → B ∞ extends to a continuous homomorphism T θ, m : A θ, m → B θ, m . If moreover T isinjective (respectively surjective) then T θ, m is injective (respectively surjective) too. Proposition . Fix m ∈ Θ ( B ) and let ( A, α ) be a C ∗ -algebra endowed witha strongly continuous and isometric action of B and let also I be an α -invariant(essential) ideal of A . Then I θ, m is an (essential) ideal of A θ, m . K -theory In this final section, we show that the K -theory is an invariant of our C ∗ -deformation, exactly as in the Abelian case [ ]. We still consider a C ∗ -algebra A ,endowed with a strongly continuous and isometric action α of a normal j -group B . .7. INVARIANCE OF THE K -THEORY 141 Given an element m ∈ Θ ( B ), we form A θ, m , the C ∗ -deformation of A . We endow B θ, m ( B , A ) with the action ˆ α of B defined on B ( B , A ) by (cid:0) ˆ α g F (cid:1) ( x ) := α g (cid:0) F ( g − x ) (cid:1) . (7.21)We will show that ˆ α is a proper action of B on the C ∗ -subalgebra S θ, m ( B , A ), andthat A θ, m is the generalized fixed point algebra for this action, in the sense of [ ].We also let ˜ α denote the extension of α from A to B ( B , A ), given by (cid:0) ˜ α g ( F ) (cid:1) ( x ) = α g (cid:0) F ( x ) (cid:1) , so that ˆ α g = ˜ α g ◦ L ⋆g = L ⋆g ◦ ˜ α g , ∀ g ∈ B . This also shows that the infinitesimal form of ˜ α on B ( B , A ∞ ) is related with theinfinitesimal form of α on A ∞ by (cid:0) X ˜ α f (cid:1) ( x ) = X α (cid:0) f ( x ) (cid:1) , ∀ X ∈ U ( b ) , ∀ f ∈ B ( B , A ∞ ) . We start with some preliminary results:
Lemma . The action ˆ α is isometric on B θ, m ( B , A ) and its restriction to S θ, m ( B , A ) is strongly continuous. Proof.
The isometry follows from the covariance of the pseudo-differentialcalculus: for all g ∈ B , we haveΩ θ, m (cid:0) ˆ α g ( F ) (cid:1) = Id B ( H χ ) ⊗ α g (cid:0) Ω θ, m ( L ⋆g F ) (cid:1) = Id B ( H χ ) ⊗ α g (cid:0) U θ ( g ) ◦ Ω θ, m ( F ) ◦ U θ ( g ) ∗ (cid:1) , ∀ F ∈ B ( B , A ) . For the strong continuity of ˆ α on S θ, m ( B , A ), it suffices (by density) to prove it on S S can ( B , A ). Since by Theorem 7.20 the operator norm is weaker than the Fr´echettopology of B ( B , A ), it suffices to prove strong continuity on S S can ( B , A ) for theinduced Fr´echet topology of B ( B , A ). As S S can ( B , A ) is a subset of B ( B , A ) stableunder the inversion map of B and that the right regular action R ⋆ is strongly con-tinuous on B ( B , A ), we deduce that the left regular action L ⋆ is strongly continuouson S S can ( B , A ). Since ˆ α = ˜ α ◦ L ⋆ = L ⋆ ◦ ˜ α , it then remains to prove strong continuityof ˜ α on S S can ( B , A ) for the induced Fr´echet topology of B ( B , A ). As ˜ α commuteswith the right regular action R ⋆ , it commutes with the left-invariant differentialoperators. By isometry of α , strong continuity everywhere will follow from strongcontinuity at the neutral element e B . This also implies that for f ∈ S S can ( B , A ), wehave in the semi-norms defining the topology of B ( B , A ): k ˜ α g ( f ) − f k k = sup X ∈ U k ( b ) k e X (˜ α g f ) − e Xf k ∞ | X | k = sup X ∈ U k ( b ) k ˜ α g ( e Xf ) − e Xf k ∞ | X | k , ∀ g ∈ B . Hence it suffices to consider the case k = 0. By a compactness argument, we seethat the strong continuity of α on A implies strong continuity of ˜ α on D ( B , A ) forthe uniform norm, namely k ˜ α g ( f ) − f k ∞ → g → e B and one concludes by densityof D ( B , A ) in S S can ( B , A ). (cid:3)
42 7. DEFORMATION OF C ∗ -ALGEBRAS We now come to the question of the properness of the action ˆ α on S θ, m ( B , A ),in the sense of [ ]. For this property to hold, we need to find a dense ˆ α -invariant ∗ -subalgebra B of S θ, m ( B , A ), such that for all f , f ∈ B , the maps (cid:2) g ∈ B ∆ B ( g ) − / f ⋆ θ, m ˆ α g ( f ∗ θ, m ) (cid:3) and (cid:2) g ∈ B f ⋆ θ, m ˆ α g ( f ∗ θ, m ) (cid:3) , belong to L (cid:0) B , S θ, m ( B , A ) (cid:1) and denoting by M ( B ) ˆ α the subalgebra of the multi-plier algebra of S θ, m ( B , A ) of elements which preserve B and which are invariantunder the extension of ˆ α to M (cid:0) S θ, m ( B , A ) (cid:1) we have h f ∈ B Z B f ⋆ θ, m ˆ α g ( f ∗ θ, m ⋆ θ, m f ) d B ( g ) i ∈ M ( B ) ˆ α . Our candidate for B is S S can ( B , A ∞ ). Note first: Lemma . Let m ∈ Θ ( B ) . Then (cid:0) S S can ( B , A ∞ ) , ⋆ θ, m (cid:1) is a dense ˆ α -invariant ∗ -subalgebra of S θ, m ( B , A ) . Proof.
That S S can ( B , A ∞ ) is closed under the involution ∗ θ, m and under theaction ˆ α of B is clear. Since S S can ( B , A ∞ ) is dense in S S can ( B , A ) for the Fr´echettopology of the latter, since S S can ( B , A ) is dense on S θ, m ( B , A ) for the C ∗ -topologyof the latter and since the C ∗ -topology is weaker than the Fr´echet topology on S S can ( B , A ), the density statement follows. Last, that (cid:0) S S can ( B , A ∞ ) , ⋆ θ, m (cid:1) is asubalgebra of S θ, m ( B , A ) follows from Proposition 3.10. (cid:3) Lemma . Let f , f ∈ S S can ( B , A ∞ ) . Then the map ( x , x ) ∈ B × B (7.22) h y ∈ B R ⋆x ( f ) R ⋆x (cid:0) ˆ α y ( f ) (cid:1) = (cid:2) z ∈ B f ( zx ) α y (cid:0) f ( y − zx ) (cid:1) ∈ A (cid:3)i , belongs to B µ (cid:0) B × B , S S can (cid:0) B , S S can ( B , A ∞ ) (cid:1)(cid:1) , with µ := { d n ( j ,k ; j ,k ; m ) B × B } ( j ,k ; j ,k ; m ) ∈ N , where ( j , k ; j , k ; m ) ∈ N labels the semi-norms of S S can (cid:0) B , S S can ( B , A ∞ ) (cid:1) and n ( j , k ; j , k ; m ) ∈ N is linear in its arguments. Proof.
Recall that the Fr´echet topology of S S can ( B , A ∞ ) can be associatedwith the set of semi-norms k f k ( j,k ) ,m := sup X ∈ U k ( b ) sup Y ∈ U m ( b ) sup x ∈ B d B ( x ) j k Y α e X f ( x ) k| X | k | Y | m , and the one of B µ (cid:0) B × B , S S can (cid:0) B , S S can ( B , A ∞ ) (cid:1)(cid:1) can be associated with the set ofsemi-norms k F k l, ( j ,k ) , ( j ,k ) ,m := sup X ∈ U l ( b ⊕ b ) sup Y ∈ U k ( b ) sup Y ∈ U k ( b ) sup Z ∈ U m ( b ) sup x ,x ,y,z ∈ B d B ( y ) j d B ( z ) j (cid:13)(cid:13) Z α e Y ,z e Y ,y e X ( x ,x ) F ( x , x ; y ; z ) (cid:3)(cid:1)(cid:13)(cid:13) d B × B ( x , x ) n ( j ,k ,j ,k ,m ) | X | l | Y | k | Y | k | Z | m . .7. INVARIANCE OF THE K -THEORY 143 Applying this to the four-point function constructed in (7.22), we get using theSweedler notation: Z α e Y ,z e Y ,y e X ( x ,x ) h f ( zx ) α y (cid:0) f ( y − zx ) (cid:1)i = X ( X ) Z α e Y ,z e Y ,y h(cid:0) f X f (cid:1) ( zx ) α y (cid:0)(cid:0) f X f (cid:1) ( y − zx ) (cid:1)i = X ( X ) X ( Y ) Z α e Y ,z h(cid:0) f X f (cid:1) ( zx ) α y (cid:0)(cid:0) Y α , Y , f X f (cid:1) ( y − zx ) (cid:1)i = X ( X ) X ( Y ) X ( Y ) Z α h(cid:0) ^ Ad x − Y , f X f (cid:1) ( zx ) α y (cid:0)(cid:0) ^ Ad x − Y , Y α , Y , f X f (cid:1) ( y − zx ) (cid:1)i = X ( X ) X ( Y ) X ( Y ) X ( Z ) (cid:0) Z α ^ Ad x − Y , f X f (cid:1) ( zx ) α y (cid:0)(cid:0) ( Ad y − Z ) α ^ Ad x − Y , Y α , Y , f X f (cid:1) ( y − zx ) (cid:1) . Hence we getsup X ∈ U l ( b ⊕ b ) sup Y ∈ U k ( b ) sup Y ∈ U k ( b ) sup Z ∈ U m ( b ) (cid:13)(cid:13)(cid:13) Z α e Y ,z e Y ,y e X ( x ,x ) h f ( zx ) α y (cid:0) f ( y − zx ) (cid:1)i(cid:13)(cid:13)(cid:13) | X | l | Y | k | Y | k | Z | m ≤ C ( l, k , k , m ) d B × B ( x , x ) k d B ( y ) m d B ( zx ) N d B ( y − zx ) M k f k ( N,l + k ) ,m k f k ( M,l + k + k ) ,k + m , with N, M ∈ N arbitrary. This implies that the norm k . k l, ( j,k ) of the map (7.22) issmaller than a constant times: d B ( y ) j d B ( z ) j d B × B ( x , x ) k d B ( y ) m d B × B ( x , x ) n ( j ,k ,j ,k ,m ) d B ( zx ) N d B ( y − zx ) M × k f k ( N,l + k ) ,m k f k ( M,l + k + k ) ,k + m . As M, N are arbitrary, using the sub-multiplicativity of the modular weight and thefact that d B × B ≍ d B ⊗ ⊗ d B (proved in Lemma 1.5), we can find n ( j , k , j , k , m )such that the fraction above is uniformly bounded. This achieves the proof. (cid:3) Proposition . For m ∈ Θ ( B ) and f , f ∈ S S can ( B , A ∞ ) , the maps (cid:2) g ∈ B ∆ B ( g ) − / f ⋆ θ, m ˆ α g ( f ∗ θ, m ) (cid:3) and (cid:2) g ∈ B f ⋆ θ, m ˆ α g ( f ∗ θ, m ) (cid:3) , belong to L (cid:0) B , S θ, m ( B , A ) (cid:1) . Moreover, the map Λ f ,f : S S can ( B , A ∞ ) → S S can ( B , A ∞ ) , f Z B f ⋆ θ, m ˆ α g ( f ∗ θ, m ⋆ θ, m f ) d B ( g ) , belongs to M (cid:0) S S can ( B , A ∞ ) (cid:1) ˆ α , the subalgebra of the multiplier algebra of S θ, m ( B , A ) consisting of ˆ α -invariant and S S can ( B , A ∞ ) -preserving elements. Consequently, theaction ˆ α of B on S θ, m ( B , A ) is proper in the sense of [ ]. Proof.
The first part of the claim follows from the definition of the product ⋆ θ, m in term of an oscillatory integral: f ⋆ θ, m ˆ α g ( f ∗ θ, m ) = ^ Z B × B K θ, m ( x , x ) R ⋆x ( f ) R ⋆x (cid:0) ˆ α g ( f ∗ θ, m ) (cid:1) d B ( x ) d B ( x ) ,
44 7. DEFORMATION OF C ∗ -ALGEBRAS combined with Lemma 7.44 and with the inclusion S S can (cid:0) B , S S can ( B , A ∞ ) (cid:1) ⊂ L (cid:0) B , S θ, m ( B , A ) (cid:1) , together with the fact that the multiplication by ∆ − / B is continuous on the space S S can (cid:0) B , S θ, m ( B , A ) (cid:1) as the latter is a tempered function. For the second part,we again use Lemma 7.44 which shows (since S S can ( B , A ∞ ) is an algebra for theproduct ⋆ θ, m ) that the map Λ f ,f sends continuously S S can ( B , A ∞ ) to itself. Sincethe ˆ α -invariance of the map above is rather clear, it remains to show that the latterextends from S θ, m ( B , A ) to itself, i.e . that it is indeed an element of the multiplieralgebra of S θ, m ( B , A ). For this we need to find a convenient expression for Z B f ⋆ θ, m ˆ α g ( f ) d B ( g ) , ∀ f , f ∈ S ( B , A ∞ ) , ∀ y ∈ B . By Proposition 1.36, we can express the oscillatory integral underlying the product ⋆ θ, m in term of absolutely convergent integral. More precisely, writing K θ, m = e iθ S can A θ, m , we have Z B f ⋆ θ, m ˆ α g ( f )( y ) d B ( g ) = Z B e iθ S can ( x ,x ) D x ,x h A m ( x , x ) f ( yx ) α g (cid:0) f ( g − yx ) (cid:1)i d B ( g ) d B ( x ) d B ( x ) , where one can choose the operator D such that the coefficients decay (in a temperedway) as fast as one wishes, so that taking into account the decay of f , f , the triple-integral above is absolutely convergent. As D commutes with left translations, weget after the change of variable g yx g : Z B f ⋆ θ, m ˆ α g ( f )( y ) d B ( g ) = Z B e iθ S can ( x ,x ) D x ,x h A m ( x , x ) f ( yx ) α yx g (cid:0) f ( g − ) (cid:1)i d B ( g ) d B ( x ) d B ( x ) . By (the ordinary) Fubini Theorem, the triple integral above becomes: Z B e iθ S can ( x ,x ) D x ,x h A θ, m ( x , x ) f ( yx ) α yx (cid:16) Z B α g (cid:0) f ( g − ) (cid:1) d B ( g ) (cid:17)i × d B ( x ) d B ( x ) , which means that as expected, we can write Z B f ⋆ θ, m ˆ α g ( f ) d B ( g ) = f ⋆ θ, m α (cid:16) Z B α g (cid:0) f ( g − ) d B ( g ) (cid:1)(cid:17) . Now, we observes that the map(7.23) f Z B α g (cid:0) f ( g − ) (cid:1) d B ( g ) , sends continuously S S can ( B , A ∞ ) to A ∞ . Indeed, for X ∈ U ( b ), we have X α Z B α g (cid:0) f ( g − ) (cid:1) d B ( g ) = Z B α g (cid:0) ( Ad g − X ) α (cid:0) f ( g − ) (cid:1)(cid:1) d B ( g ) , which entails that (cid:13)(cid:13)(cid:13) X α Z B α g (cid:0) f ( g − ) (cid:1) d B ( g ) (cid:13)(cid:13)(cid:13) ≤ Z B d B ( g ) deg( X ) (cid:13)(cid:13) X α (cid:0) f ( g − ) (cid:1)(cid:13)(cid:13) d B ( g ) , .7. INVARIANCE OF THE K -THEORY 145 which is finite since the group inverse map is continuous on S S can ( B , A ). This isenough to conclude in view of Lemma 7.24. (cid:3) The C ∗ -subalgebra of M (cid:0) S θ, m ( B , A )) generated by the operations described inthe previous Proposition is called the generalized fixed point algebra of S θ, m ( B , A )and is denoted symbolically by S θ, m ( B , A ) ˆ α . Also, it is proved in [ ] that thelinear span of (cid:8)(cid:2) g ∈ B ∆( g ) − / f ⋆ θ, m ˆ α g ( f ∗ θ, m ) ∈ S S can ( B , A ∞ ) (cid:3) :(7.24) f , f ∈ S S can ( B , A ∞ ) (cid:9) , forms a subalgebra of the crossed product B⋉ ˆ α S θ, m ( B , A ), whose closure is stronglyMorita equivalent to the generalized fixed point algebra, with equivalence bi-modulegiven by S θ, m ( B , A ) itself. Our final task is to show that S θ, m ( B , A ) ˆ α is isomorphicto A θ, m and that the action is saturated, meaning that the algebra generated bythe set of functions (7.24) is dense in the crossed product B ⋉ ˆ α S θ, m ( B , A ). Notethat here, there is no distinction here between the reduced and full crossed productalgebras, since B is solvable and thus amenable. Lemma . With the notations as above, we have S θ, m ( B , A ) ˆ α ≃ A θ, m . Proof.
Call P : S S can ( B , A ∞ ) → A ∞ the map given in (7.23). As observedearlier, P sends S S can ( B , A ∞ ) to A ∞ ⊂ A θ, m . As S θ, m ( B , A ) ˆ α is the closure of theimage of P (cid:0) S S can ( B , A ∞ ) (cid:1) under the map α : A ∞ → B ( B , A ), in M (cid:0) S θ, m ( B , A ) (cid:1) and that B ( B , A ) ⊂ M (cid:0) S θ, m ( B , A ) (cid:1) isometrically (by Theorem 7.33), we get that S θ, m ( B , A ) ˆ α ⊂ α ( A θ, m ).Now, let ϕ j ∈ S S can ( B ) and a j ∈ A ∞ , j = 1 ,
2, so that ϕ j ⊗ a j ∈ S S can ( B , A ∞ )and P (cid:0) ϕ ⊗ a ⋆ θ, m ϕ ⊗ a (cid:1) = Z B α g ( a a ) ϕ ⋆ θ, m ϕ ( g − ) d B ( g ) . By isometry of the action α and Lemma 4.5, we therefore get for every k ∈ N : (cid:13)(cid:13) P (cid:0) ϕ ⊗ a ⋆ θ, m ϕ ⊗ a (cid:1) − Z B α g ( a a ) ϕ ( g − ) d B ( g ) (cid:13)(cid:13) k = sup X ∈ U k ( b ) | X | − k (cid:13)(cid:13)(cid:13) Z B (cid:0) X α α g ( a a ) (cid:1) (cid:0) ϕ ⋆ θ, m ϕ ( g − ) − ϕ ( g − (cid:1) d B ( g ) (cid:13)(cid:13)(cid:13) ≤ C k k a a k k Z B d B ( g ) k | ϕ ⋆ θ, m ϕ ( g − ) − ϕ ( g − ) | d B ( g ) . Now, by Lemma 2.31 we see that there exists n ∈ N such that d − n B ∈ L ( B ). Thisimplies that the quantity above is (up to a constant) bounded by: k a a k k k d − n B k k d k + n B (cid:0) ϕ ⋆ θ, m ϕ − ϕ (cid:1) k ∞ = k a a k k k d − n B k k ϕ ⋆ θ, m ϕ − ϕ k ,k + n , where the last semi-norm is the one on the one-point Schwartz space S S can ( B ),as given in Remark 2.36. By Proposition 4.19 applied to the Fr´echet algebra S S can ( B ) (which is its own subspace of smooth vectors for the right regular action)we can let ϕ range over an approximate unit for the deformed Fr´echet algebra (cid:0) S S can ( B ) , ⋆ θ, m (cid:1) . Thus, P (cid:0) ϕ ⊗ a ⋆ θ, m ϕ ⊗ a (cid:1) will converges to Z B α g ( a a ) ϕ ( g − ) d B ( g ) ,
46 7. DEFORMATION OF C ∗ -ALGEBRAS for the Fr´echet topology of A ∞ . Let then ϕ ranging over approximate δ -functionssupported at the neutral element. The latter integral will converges to a a for theFr´echet topology of A ∞ , and thus for the norm topology of A θ, m . We conclude bythe density of A ∞ .A ∞ in A θ, m .A θ, m and by the density of A θ, m .A θ, m in A θ, m . (cid:3) To prove that the action ˆ α of B on S θ, m ( B , A ) is saturated, we need an inver-sion formula for the product ⋆ θ, m . This result (for an elementary normal j -group)is extracted from [ ] but we provide the detailed arguments for the sake of com-pleteness. Proposition . Let m ∈ Θ ( B ) , let B be a normal j -group and let f , f ∈S S can ( B , A ) . Then for all z ∈ B , we have the absolutely convergent representation: Z B × B K B − θ, m m ( x , x ) (cid:0) R ⋆x − f (cid:1) ⋆ θ, m (cid:0) R ⋆x − f (cid:1) ( z ) d B ( x ) d B ( x ) = f ( z ) f ( z ) . Proof.
Fix z ∈ B and consider the continuous map S S can ( B , A ) → S S can ( B , A ) , f (cid:2) x ∈ B (cid:0) R ⋆x − f (cid:1) ( z ) = f ( zx − ) ∈ A (cid:3) . Then, using arguments very similar to those leading to Lemmas 1.42 and 1.49, wecan deduce that the map( f , f ) h ( y , y ) ∈ B × B (cid:2) ( x , x ) (cid:0) R ⋆y x − f (cid:1) ( z ) (cid:0) R ⋆y x − f (cid:1) ( z ) ∈ A (cid:3)i , is continuous from S S can ( B , A ) × S S can ( B , A ) to B µ (cid:0) B × B , S S can ( B × B , A ) (cid:1) for asuitable family µ of tempered weights on B × B . Hence, the map( f , f ) h ( x , x ) ∈ B × B (cid:0) R ⋆x − f (cid:1) ⋆ θ, m (cid:0) R ⋆x − f (cid:1) ( z ) i , is continuous from S S can ( B , A ) ×S S can ( B , A ) to S S can ( B × B , A ), which in turn entailsthat the map( f , f ) h ( x , x ) ∈ B × B K B − θ, m m ( x , x ) (cid:0) R ⋆x − f (cid:1) ⋆ θ, m (cid:0) R ⋆x − f (cid:1) ( z ) i , is continuous from S S can ( B , A ) × S S can ( B , A ) to S S can ( B × B , A ), since the kernel K B − θ, m m is tempered. In summary, the expression Z B × B K B − θ, m m ( x , x ) (cid:0) R ⋆x − f (cid:1) ⋆ θ, m (cid:0) R ⋆x − f (cid:1) ( z ) d B ( x ) d B ( x ) , is well defined as an absolutely convergent integral for all z ∈ B . Next, fromRemark 1.51, for the restriction of ⋆ θ, m to S S can ( B , A ) × S S can ( B , A ), we have the(point-wise) absolutely convergent expression: f ⋆ θ, m f = Z B × B K B θ, m ( x ′ , x ′ ) R ⋆x ′ ( f ) R ⋆x ′ ( f ) d B ( x ′ ) d B ( x ′ ) . Hence, .7. INVARIANCE OF THE K -THEORY 147 Z B × B K B − θ, m m ( x , x ) R ⋆x − ( f ) ⋆ θ, m R ⋆x − ( f ) d B ( x ) d B ( x )= Z B × B K B − θ, m m ( x , x ) (cid:16) Z B × B K B θ, m ( x ′ , x ′ ) R ⋆x ′ x − ( f ) R ⋆x ′ x − ( f ) d B ( x ′ ) d B ( x ′ ) (cid:17) × d B ( x ) d B ( x )= Z B × B K B − θ, m m ( x , x ) (cid:16) Z B × B K B θ, m ( x ′ x , x ′ x ) R ⋆x ′ ( f ) R ⋆x ′ ( f ) d B ( x ′ ) d B ( x ′ ) (cid:17) × ∆ − B ( x ) ∆ − B ( x ) d B ( x ) d B ( x ) . The rest of the proof is computational: we are going to show (in the distributionalsense and up to numerical pre-factors) that T B x ′ ,x ′ := Z B × B K B − θ, m m ( x , x ) K B θ, m ( x ′ x , x ′ x ) ∆ − B ( x ) ∆ − B ( x ) d B ( x ) d B ( x )= δ B e ( x ′ ) δ B e ( x ′ ) , where δ B g is the Dirac measure on B supported at g . We first prove that withoutloss of generality, we may assume that B is an elementary normal j -group. Indeed,let B ′ ⋉ R S be the Pyatetskii-Shapiro decomposition of B and assume that T B ′ = δ B ′ e ⊗ δ B ′ e and T S = δ S e ⊗ δ S e . As usual, an element g ∈ B is parametrized as g = g g ′ , where g ′ ∈ B ′ , g ∈ S . Firstly, we observe that under this decomposition,the modular function of B is the product of modular functions of B ′ and S . Thisfollows from the decomposition of left invariant Haar measures d B = d B ′ ⊗ d S , fromthe relation (7.2) in Remark 7.1 and the definition of the modular function:∆ B ( g ) = d B ( g − )d B ( g ) = d B ( g ′− g − )d B ( g g ′ ) = d B (cid:0) R g ′− ( g − ) g ′− (cid:1) d B ( g g ′ )= d S (cid:0) R g ′− ( g − ) (cid:1) d B ′ ( g ′− (cid:1) d S ( g )d B ′ ( g ′ ) = ( R g ′− ) ⋆ d S ( g − )d B ′ ( g ′− (cid:1) d S ( g )d B ′ ( g ′ )= d S ( g − )d B ′ ( g ′− (cid:1) d S ( g )d B ′ ( g ′ ) = ∆ S ( g )∆ B ′ ( g ′ ) . Also, by construction, we have K B θ, m ( x, y ) = K B ′ θ, m ′ ( x ′ , y ′ ) K S θ, m ( x , y ) , and thus K B θ, m ( xy, st ) = K B ′ θ, m ′ ( x ′ y ′ , s ′ t ′ ) K S θ, m (cid:0) x R x ′ ( y ) , s R s ′ ( t ) (cid:1) . Hence, for x = x x ′ , s = s s ′ , y = y y ′ , t = t t ′ ∈ B with x , s , y , t ∈ S and x ′ , s ′ , y ′ , t ′ ∈ B ′ , we get from our induction hypothesis, that the distribution T B x,s
48 7. DEFORMATION OF C ∗ -ALGEBRAS reads: Z S × B ′ K B ′ − θ, m ′ m ′ ( y ′ , t ′ ) K S − θ, m m ( y , t ) K B ′ θ, m ′ ( x ′ y ′ , s ′ t ′ ) K S θ, m (cid:0) x R x ′ ( y ) , s R s ′ ( t ) (cid:1) × ∆ − B ′ ( y ′ ) ∆ − S ( y ) ∆ − B ′ ( t ′ ) ∆ − S ( t ) d B ′ ( y ′ ) d S ( y ) d B ′ ( t ′ ) d S ( t )= T B ′ x ′ ,s ′ Z S K S − θ, m m ( y , t ) K S θ, m (cid:0) x R x ′ ( y ) , s R s ′ ( t ) (cid:1) × ∆ − S ( y ) ∆ − S ( t ) d S ( y ) d S ( t )= δ B ′ e ( x ′ ) δ B ′ e ( t ′ ) Z S K S − θ, m m ( y , t ) K S θ, m ( x y , s t ) × ∆ − S ( y ) ∆ − S ( t ) d S ( y ) d S ( t )= δ B ′ e ( x ′ ) δ B ′ e ( t ′ ) T S x ,s = δ B ′ e ( x ′ ) δ B ′ e ( t ′ ) δ S e ( x ) δ S e ( s ) = δ B e ( x ) δ B e ( t ) . Now, for B = S an elementary normal j -group, using the explicit expressionfor the two-point kernels, we find in the coordinates (2.4) and up to numericalpre-factors: T S x ′ ,x ′ = Z R d +4 d a d a d v d v d t d t × m ( a − a ) m ( − a − a ′ ) m ( a + a ′ ) m ( − a ) m ( a ) m ( a + a ′ − a − a ′ ) m ( a ) m ( a ) m ( a + a ′ − a − a ′ )∆ S ( a )∆ S ( a ) × exp n iθ (cid:16) t (cid:0) sinh 2 a − sinh 2( a + a ′ ) (cid:1) − t (cid:0) sinh 2 a − sinh 2( a + a ′ ) (cid:1)(cid:17)o × exp n iθ (cid:16) t ′ e − a sinh 2( a + a ′ ) − t ′ e − a sinh 2( a + a ′ ) (cid:17)o × exp n iθ ω ( v , v ) (cid:16) cosh a cosh a − cosh( a + a ′ ) cosh( a + a ′ ) (cid:17)o × exp n iθ (cid:16) ω ( v , v ′ ) e − a sinh 2( a + a ′ ) − ω ( v , v ′ ) e − a sinh 2( a + a ′ ) (cid:17)o × exp n iθ (cid:16) ω ( v , v ′ ) e − a cosh( a + a ′ ) cosh( a + a ′ ) (cid:17)o × exp n − iθ (cid:16) ω ( v , v ′ ) e − a cosh( a + a ′ ) cosh( a + a ′ ) (cid:17)o × exp n − iθ ω ( v ′ , v ′ ) e − a − a cosh( a + a ′ ) cosh( a + a ′ ) o , Integrating out the variables t , t , yields a factor sech2 a sech2 a δ ( a ′ ) δ ( a ′ ), andthus we get: T S x ′ ,x ′ = δ ( a ′ ) δ ( a ′ ) Z R d +2 m ( a ) m ( a ) m ( a − a )cosh 2 a cosh 2 a ∆ S ( a )∆ S ( a ) × exp n iθ (cid:16) t ′ e − a sinh 2 a − e − a t ′ sinh 2 a (cid:17)o × exp n iθ ω (cid:0) v , v ′ e − a sinh 2 a + 2 v ′ e − a cosh a cosh a (cid:1)o × exp n − iθ ω (cid:0) v , v ′ e − a sinh 2 a + 2 v ′ e − a cosh a cosh a (cid:1)o × exp n − iθ ω ( v ′ , v ′ ) e − a − a cosh a cosh a o d a d a d v d v . .7. INVARIANCE OF THE K -THEORY 149 Integrating out the variables v , v , yields a factor: δ (cid:0) v ′ e − a sinh 2 a + 2 v ′ e − a cosh a cosh a (cid:1) × δ (cid:0) v ′ e − a sinh 2 a + 2 v ′ e − a cosh a cosh a (cid:1) . Observe that the Jacobian of the map V × V → V × V , ( v ′ , v ′ ) (cid:0) v ′ e − a sinh 2 a + 2 v ′ e − a cosh a cosh a ,v ′ e − a sinh 2 a + 2 v ′ e − a cosh a cosh a (cid:1) , is proportional to e − d ( a + a ) cosh d a cosh d a cosh d ( a − a ). Thus, the former δ function is proportional to: e d ( a + a ) sech d a sech d a sech d ( a − a ) δ ( v ′ ) δ ( v ′ ) , and consequently, T S x ′ ,x ′ = δ ( a ′ ) δ ( a ′ ) δ ( v ′ ) δ ( v ′ ) × Z R e − a + a ) cosh 2( a − a )exp n iθ (cid:16) t ′ e − a sinh 2 a − t ′ e − a sinh 2 a (cid:17)o d a d a . But the pre-factor e − a + a ) cosh 2( a − a ) in the expression above is exactly theJacobian of the map: R → R , ( a , a ) (cid:0) e − a sinh 2 a , − e − a sinh 2 a (cid:1) , so that T S x ′ ,x ′ = δ ( a ′ ) δ ( a ′ ) δ ( v ′ ) δ ( v ′ ) Z R exp n iθ (cid:16) u t ′ + u t ′ (cid:17)o d u d u = δ ( a ′ ) δ ( a ′ ) δ ( v ′ ) δ ( v ′ ) δ ( t ) δ ( t ) = δ S ( x ) δ S ( x ) . This concludes the proof. (cid:3)
Lemma . The action ˆ α of B on S θ, m ( B , A ) is saturated, that is, the con-volution algebra generated by the elements (7.25) (cid:2) g ∈ B ∆( g ) − / f ⋆ θ, m ˆ α g ( f ∗ θ, m ) ∈ S S can ( B , A ∞ ) (cid:3) , where f , f ∈ S S can ( B , A ∞ ) , is dense in the crossed product C ∗ -algebra B ⋉ ˆ α S θ, m ( B , A ) . Proof.
Call E the subalgebra of B ⋉ ˆ α S θ, m ( B , A ) generated by the elementsgiven in (7.25) and E its closure for the C ∗ -norm of the crossed product algebra.We need to show that E = B ⋉ ˆ α S θ, m ( B , A ). By Lemma 7.44, we know that E ⊂ S S can (cid:0) B , S S can ( B , A ∞ ) (cid:1) ⊂ L (cid:0) B , S θ, m ( B , A ) (cid:1) , so that E ⊂ B ⋉ ˆ α S θ, m ( B , A ). Since E is a two-sided ideal of B ⋉ ˆ α S θ, m ( B , A ) [ ,Theorem 1.5], the converse inclusion will clearly follow if we prove the existence ofa bounded approximate unit of B ⋉ ˆ α S θ, m ( B , A ) consisting of elements of E . Butthis follows from the following arguments:The inversion formula of Proposition 7.47 gives in this context:∆( g ) − / Z B × B K − θ, m m ( u, v ) R ⋆u − ( f ∗ θ, m ) ⋆ θ, m R ⋆v − (cid:0) ˆ α g ( f ) (cid:1) d B ( u ) d B ( v ) =∆( g ) − / f ∗ θ, m ˆ α g ( f ) ,
50 7. DEFORMATION OF C ∗ -ALGEBRAS point-wise for any f , f ∈ S S can ( B , A ∞ ). Since the right regular action R ⋆ preserves S S can ( B , A ∞ ) and commutes with the action ˆ α given in (7.21), approximating theRiemann integral above by Riemann sums, shows that the maps (cid:2) g ∈ B ∆( g ) − / f ∗ θ, m ˆ α g ( f ) ∈ S ( B , A ∞ ) (cid:3) , ∀ f , f ∈ S ( B , A ∞ ) , belong to E as well. But with such maps, it is easy to construct an approximateunit in B ⋉ ˆ α S θ, m ( B , A ). Indeed, as Z B R ⋆u ( δ B ) L ⋆g R ⋆u ( δ B ) d B ( u ) = δ B ( g ) ⊗ , where 1 above is the constant unit function of B ( B ), it suffices to consider the net ofelements ∆( g ) − / ϕ λ ⊗ a λ ˆ α g ( ϕ λ ⊗ a λ ), where ϕ λ are in S S can ( B ) and approximatethe Dirac measure supported at the neutral element, and where a λ is a boundedapproximate unit for A ∞ . (cid:3) Corollary . The deformed C ∗ -algebra A θ, m is strongly Morita equivalentto the reduced crossed product B ⋉ ˆ α S θ, m ( B , A ) . Proof.
By [ , Theorem 1.5], E (as described above) is strongly Morita equiv-alent to the generalized fixed point algebra S θ, m ( B , A ) ˆ α for the action ˆ α . Then oneconcludes using Lemmas 7.46 and 7.48. (cid:3) We are now able to prove the main result of this section:
Theorem . For all m ∈ Θ ( B ) and θ ∈ R ∗ , we have K ∗ ( A θ, m ) ≃ K ∗ ( A ) , ∗ = 0 , . Proof.
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