On the degenerate principal series of complex symplectic groups
aa r X i v : . [ m a t h . R T ] F e b ON THE DEGENERATE PRINCIPAL SERIES OFCOMPLEX SYMPLECTIC GROUPS
PIERRE CLARE
Abstract.
We apply techniques introduced by Clerc, Kobayashi, Ørstedand Pevzner to study the degenerate principal series of
Sp( n, C ) . An ex-plicit description of the K -types is provided and Knapp-Stein normalisedoperators are realised as symplectic Fourier transforms, and their K -spectrum explicitely computed. Reducibility phenomena are analysedin terms of K -types and eigenvalues of intertwining operators. We alsoconstruct a new model for these representations, in which Knapp-Steinintertwiners take an algebraic form. Introduction
Background and purpose.
Among the representations of reductive Liegroups, the so-called small representations have received a lot of attentionfor several years. After the accomplishments of the 90’s, mainly obtainedthrough algebraic methods (see the introduction and the references in [8]),recent progress has stemmed from the developement of new techniques ingeometric analysis. Leading work in this direction was the series [9, 10,11], dealing with the minimal representation of general indefinite orthogonalgroups.More recently, new techniques were introduced in this area by J.-L. Clerc,T. Kobayashi, B. Ørsted and M. Pevzner in relation with various problemsin classical analysis [2] and representation theory [12]. The first of theseworks is devoted to the computation of certain triple integrals, and featuresthe study of certain intertwining operators related to representations of thereal symplectic group. More precisely, it is observed that some Knapp-Steinoperators can be realised as Fourier transforms once normalised. This ideawas already used in the work of A. Unterberger in the case of
SL(2 , R ) , see[16]. It was extended to SL( n, R ) in [15] and to Sp( n, R ) for the first time in[13]. It is taken further in [12] where, among many other results, a completedescription of Knapp-Stein intertwiners between the degenerate principalseries of Sp( n, R ) is carried out. This point of view allows to establish veryprecise statements, such as K -type formulas and explicit computations of Mathematics Subject Classification.
Key words and phrases.
Small representations, principal series, symplectic groups,Knapp-Stein operators, branching laws.This work was primarily supported by a JSPS postdoctoral fellowship. Support wasalso provided by the MAPMO and the Pennsylvania State University. the K -spectrum of intertwining operators. Another feature of [12] is theconstruction of what is named there the non-standard model for degenerateprincipal series. As a first application, this apparently new picture allows todefine Knapp-Stein operators by a rather simple algebraic formula.The present article aims at adapting those techniques to the case of thecomplex symplectic group. Among the parabolically induced representationsof Sp( n, C ) , those coming from maximal parabolic subgroups may be con-sidered as the most degenerate, since functions on the corresponding flagmanifold depend on the smallest possible set of parameters. These represen-tations have first been investigated by K. I. Gross in [4].Most of our arguments are rather directly inspired by the results in [2]and [12], dealing with real symplectic groups. However, we tried to providea self-contained and detailed presentation of these techniques while adaptingthem to the complex case. Moreover, the isotypic decomposition of thequaternionic orthogonal group action on square-integrable functions over theunit sphere, studied in Section 3.1, turns out to be slightly more delicate thanthe real and complex ones. It is also worth noticing Theorem 1 accountsmore precisely for the reducibility phenomena than the original descriptionin Gross’ paper [4]. In particular, the characterisations by K -types andeigenspaces of the algebraic intertwiners in the non-standard model bothseem new.Let us finally mention that the study of explicit Knapp-Stein intertwinersas geometric transforms and the computation of their K -spectrum such asthe one provided in Proposition 3 are a current object of interest. Indeed,analogous results were recently obtained in [14] for special linear groups,using techniques of [1]. Outline.
The article is organised as follows: general notations are fixed andelementary facts regarding degenerate principal series of the complex sym-plectic groups are stated in Section 1. In Section 2, we introduce certainFourier transforms, establish some of their elementary properties and usethem to normalise Knapp-Stein operators in Proposition 1. In Section 3, westudy the branching law of the degenerate principal series representationsof
Sp( n, C ) with respect to the maximal compact subgroup K = Sp( n ) : wedescribe K -types in Proposition 2 and compute the eigenvalues of Knapp-Stein operators in Proposition 3. As a result, we are able to analyse thereducible elements in the degenerate principal series in terms of K -types andeigenspaces of te Knapp-Stein intertwiners in Theorem 1. Finally, Section 4is devoted to the description of the non-standard model of the degenerateprincipal series in the sense of [12]. The main result of this section is thecomputation of the normalised Knapp-Stein operators in this picture: The-orem 2 establishes that the intertwiners are defined by an algebraic formulain this setting. EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) Setting and notations
The complex symplectic group.
For any integer p ≥ , let I p be theidentity matrix of size p and let brackets denote the associate bilinear formon C p : h X, Y i = p X k =1 x k y k for X = ( x , . . . , x p ) and Y = ( y , . . . , y p ) .Throughout this article, n shall be a fixed positive integer and N = 2 n , sothat a vector X in C N ≃ C n × C n can naturally be written X = ( X , X ) .The complex symplectic form on C N is defined by ω n ( X, Y ) = h X , Y i − h X , Y i , that is ω n ( X, Y ) = h X, J Y i where J = (cid:20) − I n I n (cid:21) . Abusing notations, we will usually drop the subscript indicating the dimen-sion and write ω ≡ ω n when no confusion may result.By definition, the complex symplectic group is the group of complex in-vertible matrices preserving ω : Sp( n, C ) = (cid:8) g ∈ GL( N, C ) (cid:12)(cid:12) ∀ X, Y ∈ C N , ω ( gX, gY ) = ω ( X, Y ) (cid:9) . Equivalently,
Sp( n, C ) is the subgroup of elements g ∈ GL( N, C ) subject tothe relation t gJ g = J .From now on, G will denote the complex symplectic group defined above.Restricting the usual Cartan involution of GL( N, C ) yields a Cartan involu-tion of G . As a consequence, K = U( N ) ∩ Sp( n, C ) is a maximal compactsubgroup of G , also called the compact symplectic group and denoted by Sp( n ) .1.2. Maximal parabolic subgroup of Heisenberg type.
Let us recallsome facts regarding complex Heisenberg groups. Let m = n − and consider H m +1 C = (cid:8) ( s, X ) ∈ C × C m (cid:9) equipped with the product ( s, X )( s ′ , X ′ ) = (cid:18) s + s ′ + 12 ω ( X, X ′ ) , X + X ′ (cid:19) , where ω ≡ ω m denotes the complex symplectic form on C m .The group G acts naturally on C N by linear applications, hence also on thecomplex projective space P N − C . The stabiliser in G of a point in P N − C is a maximal parabolic subgroup P with Langlands decomposition P = M A ¯ N ≃ (cid:0) C × . Sp( m, C ) (cid:1) ⋉ H m +1 C . PIERRE CLARE
Elements in the Cartan-stable Levi component L = M A are of the form l ( a, S ) = a s s a − s s with a ∈ C × and S = (cid:20) s s s s (cid:21) ∈ Sp( m, C ) .The Lie algebra g of G then admits a Gelfand-Naimark decomposition g = n + m + a + ¯ n and the analytic subgroup N of G with Lie algebra n isanother copy of H m +1 C which embeds in G via (1) ( s, ( X , X )) X I m s t X − t X X I m Degenerate principal series.
For a ∈ C × , we denote [ a ] = a | a | . Let ( µ, δ ) ∈ C × Z . Such a couple defines a character χ µ,δ of P by χ µ,δ ( l ( a, S )) = | a | µ [ a ] δ . From now on, we assume µ to be purely imaginary, so that χ µ,δ is unitary. Definition 1.
The induced representation π µ,δ = Ind GP χ µ,δ ⊗ is called a degenerate principal series representation of G .These representations may be described in several ways.1.3.1. Induced picture.
In this model, π µ,δ is realised on a space of square-integrable sections of the line bundle G × χ µ,δ C over the flag manifold G/P ≃ P N − C . A dense subspace of the carrying space in this picture is V ∞ µ,δ = n f ∈ C ∞ ( C N \ { } ) (cid:12)(cid:12)(cid:12) ∀ a ∈ C × , f ( a · ) = | a | − µ − N [ a ] − δ f o . By homogeneity, functions in V ∞ µ,δ are determined by their restriction to theunit sphere in C N , which we shall always identify to the N − -dimensionalEuclidean sphere S N − . The space V µ,δ is defined as the completion of V ∞ µ,δ with respect to the L -norm on S N − , and G acts by left multiplications.1.3.2. Compact picture.
Besides considering sections over the flag manifold
G/P , one may restrict the induced picture to sections over K/ ( L ∩ K ) where L denotes the Levi component C × . Sp( m, C ) of P . Considering the compactsymplectic group as the orthogonal group of a quaternionic vector space givesthe identification(2) S N − ≃ Sp( n ) / Sp( m ) EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) so that K/ ( L ∩ K ) ≃ Sp( n ) / U(1) . Sp( m ) ≃ S N − / U(1) and π µ,δ is realised on L (cid:0) S N − (cid:1) δ = n f ∈ L (cid:0) S N − (cid:1) (cid:12)(cid:12)(cid:12) ∀ θ ∈ R , f ( e iθ · ) = e − iδθ f o . The action of G in this picture is slightly more complicated than in theinduced one. However, its restriction to K reduces to the regular actionby left multiplication: π compact µ,δ ( k )( f ) = f ( k − · ) . More details about real,complex and quaternionic spheres and the isotypical decompositions of theassociated L -spaces will appear in Section 3, in order to analyse the K -typesof the representations π µ,δ and determine the behaviour of the Knapp-Steinintertwiners on these K -types.1.3.3. Non-compact picture.
Another standard picture for principal seriesrepresentations is obtained by restricting functions in the induced picture to N . More precisely, through the embedding (1), any f in V ∞ µ,δ gives a functionon H m +1 C defined by ( s, X , X ) f (1 , s, X , X ) and still denoted by f . It follows that π µ,δ is realised in L (cid:0) H m +1 C (cid:1) .In Section 4, we introduce a new model for degenerate principal seriesrepresentations and discuss its advantages in their study.2. Fourier transforms and Knapp-Stein integrals
Various integral transforms will be used in relation to Knapp-Stein in-tertwining integrals. We shall define them on the space S ( C N ) of rapidlydecreasing functions and extend them to the Schwartz space S ′ ( C N ) of tem-pered distributions by duality. The complex Fourier transform of f ∈ S ( C N ) is defined by F C N f ( ξ ) = Z C N f ( X ) e − iπ Re h X,ξ i dX. The factors in the product decomposition of C N will be labelled C ni with i ∈ { , } so that C N = C n × C n and the partial Fourier transform withrespect to the i -th variable will be denoted by F C ni . Thus for f ∈ S ( C N ) , F C n f ( X , ξ ) = Z C n f ( X , X ) e − iπ Re h X ,ξ i dX . Finally, the complex symplectic Fourier transform is defined on S ( C N ) by F symp f ( ξ ) = Z C N f ( X ) e − iπ Re ω ( X,ξ ) dX, that is F symp f ( ξ ) = F C N f ( J ξ ) . The following lemmas state elementary properties of the above transforms,to be used further.
PIERRE CLARE
Lemma 1.
Let f ∈ V − µ, − δ and a ∈ C × . Then ∀ ( X , ξ ) ∈ C N , F C n f ( aX , a − ξ ) = | a | µ [ a ] δ F C n f ( X , ξ ) Proof.
By definition, F C n f ( aX , a − ξ ) = Z C n f ( aX , X ) e − iπ Re h X ,a − ξ i dX = | a | n Z C n f ( aX , aX ) e − iπ Re h X ,ξ i dX = | a | n | a | µ − n [ a ] δ F C n f ( X , ξ ) , hence the result. (cid:3) Lemma 2.
Let f ∈ S ( C N ) . Then ∀ ( u, v ) ∈ C n × C n ≃ C N , (cid:16) F C n ◦ F symp ◦ F − C n (cid:17) f ( u, v ) = f ( v, u ) . Proof.
Let us compute: (cid:16) F C n ◦ F symp ◦ F − C n (cid:17) f ( u, v )= Z C n × C n × C n × C n f ( u ′′ , v ′′′ ) e − iπ Re ( h v ′ ,v i + h v ′′ ,u i−h u ′′ ,v ′ i−h v ′′′ ,v ′′ i ) dv ′′′ du ′′ dv ′′ dv ′ = Z C n × C n × C n × C n f ( u ′′ , v ′′′ ) e − iπ Re ( h v ′ ,v − u ′′ i + h v ′′ ,u − v ′′′ i ) dv ′′′ du ′′ dv ′′ dv ′ = Z C n × C n f ( u ′′ , v ′′′ ) δ (cid:0) v − u ′′ (cid:1) δ (cid:0) u − v ′′′ (cid:1) dv ′′′ du ′′ , hence the expected equality. (cid:3) Let us introduce some more notations: ε denote the real matrix of size N defined by blocks as follows: ε = (cid:20) I N − I N (cid:21) , and if f is a function on R N , we denote by f ε the function X f ( εX ) . Remark 1.
Under the identification between R N × R N and C N given by ( X , X ) X + iX , the transformation ε induces the complex conju-gation. It follows that the complex Fourier transform can be seen as thetransform F ε defined on S ( R N ) by: F ε f ( ξ ) = Z R N × R N f ( X ) e − iπ ( h X ,ξ i−h X ,ξ i ) d ( X , X ) , so that F ε f = ( F R N f ) ε . Indeed, if ξ = ( ξ , ξ ) and ζ = ξ + iξ , it is clearthat F C N f ( ζ ) = F ε f ( ξ ) . EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) Definition 2. If p is a function in C ∞ c (S N − ) and λ is a complex number,we denote p λ the function defined by extending p to R N \ { } by p λ ( rX ) = r λ p ( X ) for r > and X ∈ S N − .Following [2], we also consider the meromorphic function in the complexvariable λ defined by B N ( λ, k ) = π − λ − N i − k Γ (cid:0) N + k + λ (cid:1) Γ (cid:0) k − λ (cid:1) . Then, denoting H k ( R N ) the space of harmonic homogeneous polynomialsof degree k over R N , the following holds: Lemma 3.
Let p be the restriction to S N − of a polynomial in H k ( R N ) .The following identity between distributions on R N depending meromorphi-cally on λ holds: ( † ) F ε p λ = B N ( λ, k ) p ε − λ − N . Proof.
Following the lines of the proof of [2, Lemma 2.7], it is enough toprove that(3) hF ε p λ , gq i = B N ( λ, l ) h p ε − λ − N , gq i for g ∈ C ∞ c ( R + ) and q ∈ H l ( R N ) in the domain − N < Re λ < − (cid:18) N + 12 (cid:19) to ensure that ( † ) holds on R N . Local integrability of p λ and p ε − λ − N followsfrom the choice of the domain. Denoting by J µ ( ν ) the Bessel function of thefirst kind, the Bochner identity directly implies that Z S N − q ( ω ) e − i h ω,εη i dσ ( ω ) = (2 π ) N i − l ν − N J l + N − ( ν ) q ( εη ) so that F ε gq ( rω ) = Z + ∞ Z S N − g ( s ) q ( ω ′ ) e − iπrs h ω ′ ,εω i s N − ds dσ ( ω ′ )= 2 πi − l r − N q ( εω ) Z + ∞ s N g ( s ) J l + N − (2 πrs ) ds. It follows that h p λ , F ε ( gq ) i = Z + ∞ Z S N − r λ p ( ω ) F ε gq ( rω ) r N − dr dσ ( ω )= Z + ∞ Z S N − (cid:18)Z + ∞ I ( r, s ) ds (cid:19) p ( ω ) q ( εω ) dσ ( ω ) dr where we set I ( r, s ) = 2 πi − l r λ + N s N g ( s ) J l + N − (2 πrs ) . PIERRE CLARE
The proof of Claim 2.9 in [2] ensures that I belongs to L ( R + × R + , dr ds ) and that Z + ∞ I ( r, s ) dr = B N ( λ, l ) g ( s ) s − λ − . As a consequence, h p λ , F ε ( gq ) i = Z S N − p ( ω ) q ( εω ) dσ ( ω ) Z + ∞ Z + ∞ I ( r, s ) dr ds = B N ( λ, l ) Z S N − p ( ω ) q ( εω ) dσ ( ω ) Z + ∞ g ( s ) s − λ − ds, which implies that h p λ , F ε ( gq ) i = B N ( λ, l ) h p − λ − N , gq ε i , thus proving (3) and the lemma. (cid:3) Remark 2.
The link between Fourier transforms and Knapp-Stein operatorsrelies on the observation that F symp provides a unitary equivalence between π µ,δ and π − µ, − δ . Indeed, denoting f a = f ( a · ) for f ∈ S ( C N ) and a ∈ C × , asingle change of variables leads to ( F symp f ) a = | a | − N F symp ( f a − ) . If moreover f ∈ V − µ, − δ then, by linearity, ( F symp f ) a = | a | − N | a | − µ + N [ a ] − δ F symp f, that is F symp : V − µ, − δ −→ V µ,δ . Finally, for any g ∈ G , F symp ( π − µ, − δ ( g ) f ) ( ξ ) = Z C N f ( X ) e − iπ Re ω ( gX,ξ ) dX and since ω is preserved by G , it follows that F symp π − µ, − δ ( g ) = π µ,δ ( g ) F symp .Next we introduce the Knapp-Stein operators which will be related to F symp after normalisation. Definition 3.
The operator T µ,δ : V − µ, − δ −→ V µ,δ obtained by meromorphiccontinuation with respect to µ of the integral T µ,δ f ( Y ) = Z S N − f ( X ) | Re ω ( X, Y ) | − µ − N [Re ω ( X, Y )] − δ dσ ( X ) , where dσ is the Euclidean measure on the unit sphere of R N , is calledthe Knapp-Stein operator , as introduced in [6], associated to the parameter ( µ, δ ) ∈ C × Z . Remark 3.
The kernel defining the operator in the above definition dependsonly on µ and the class of δ in Z / Z . EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) The operators T µ,δ enjoy the intertwining property, yet as always in Knapp-Stein theory [6, 7], they are not unitary at first. However, normalisation maybe obtained by using the symplectic Fourier transform. More precisely, let C N ( µ, δ ) = π µ + N − Γ (cid:16) − µ − N (cid:17) Γ (cid:16) µ + N (cid:17) if δ is even − iπ µ + N − Γ (cid:16) − µ − N (cid:17) Γ (cid:16) µ + N (cid:17) if δ is odd.Then the followings holds: Proposition 1.
The normalised Knapp-Stein operator associated to ( µ, δ ) is defined by e T µ,δ = 1 C N ( µ, δ ) T µ,δ . As a meromorphic extension in the complex variable µ , it satisfies ( ‡ ) e T µ,δ = F symp (cid:12)(cid:12) V − µ, − δ . As a consequence, e T µ,δ yields a unitary equivalence between π − µ, − δ and π µ,δ .Proof. Following [2, Prop. 2.13], we identify C N \ { } to R ∗ + × S N − byusing spherical coordinates ( r, X ) . Then any function in V ∞− µ, − δ is of theform h µ − N ( rX ) = r − µ − N h ( X ) for some h ∈ C ∞ (cid:0) S N − (cid:1) satisfying h ( e iθ X ) = e iδθ h ( X ) for any X ∈ S N − and θ ∈ R , that is h is a C ∞ function in L (S N − ) − δ . The sphericalcomponent h being fixed, it is enough to prove T µ,δ h µ − N = C N ( µ, δ ) F symp (cid:12)(cid:12) V − µ, − δ h µ − N for the parameter µ in a non-empty open domain. We shall work on the setdefined by Re µ > − N . On this half-plane, h µ − N is locally integrable and,defining h ε,µ − N by h ε,µ − N ( rX ) = e − πr h µ − N ( rX ) , one has lim ε → + h ε,µ − N = h µ − N in S ( C N ) , so that F symp h µ − N = lim ε → + F symp h ε,µ − N . F symp h ε,µ − N ( sY )= Z + ∞ Z S N − e − πr r µ − N e − iπrs Re ω ( X,Y ) h ( X ) dσ ( X ) r N − dr = Z S N − F R (cid:16) r µ + N − (cid:17) ( s Re ω ( X, Y ) − iε ) h ( X ) dσ ( X ) where F R denotes the usual Fourier transform on the real line. Then, usingclassical formulas relative to Fourier transforms of homogeneous distributions(see [3]), one has F symp h ε,µ − N ( sY ) = Γ( µ + N ) e − i π ( µ + N ) (2 π ) µ + N Z S N − ( s Re ω ( X, Y ) − iε ) − µ − N h ( X ) dσ ( X ) and letting ε → + yields F symp h µ − N ( sY ) = Γ( µ + N ) e − i π ( µ + N ) (2 πs ) µ + N Z S N − (Re ω ( X, Y ) − i − µ − N h ( X ) dσ ( X ) . Recall that ( x − i − µ − N = e i π ( µ + N ) (cid:18) cos (cid:18) π ( µ + N )2 (cid:19) | x | − µ − N − i sin (cid:18) π ( µ + N )2 (cid:19) | x | − µ − N sign( x ) (cid:19) = πe i π ( µ + N ) | x | − µ − N Γ (cid:16) µ + N (cid:17) Γ (cid:16) − µ − N (cid:17) − i | x | − µ − N sign( x )Γ (cid:16) µ + N (cid:17) Γ (cid:16) − µ − N (cid:17) , using Euler’s formula. The duplication formula satisfied by Γ implies that: Γ ( µ + N ) = 2 µ + N − π − Γ (cid:18) µ + N (cid:19) Γ (cid:18) µ + N (cid:19) . Finally, we notice that h is even if h µ − N belongs to V − µ, − δ with δ ∈ Z and odd otherwise, so that F symp h µ − N ( sY ) is equal to π − µ − N + (cid:16) µ + N (cid:17) Γ (cid:16) − µ − N (cid:17) s − µ − N Z S N − | Re ω ( X, Y ) | − µ − N h ( X ) dσ ( X ) in the even case, while in the odd case it amounts to π − µ − N + − i Γ (cid:16) µ + N (cid:17) Γ (cid:16) − µ − N (cid:17) s − µ − N Z S N − | Re ω ( X, Y ) | − µ − N sign(Re ω ( X, Y )) h ( X ) dσ ( X ) , thus proving ( ‡ ). The last statement then follows from Remark 2. (cid:3) For future reference, the main Fourier transforms of use in what followsare listed below.
EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) Summary of integral transforms
Complex Fourier transform : F C N f ( ξ ) = Z C N f ( X ) e − iπ Re h X,ξ i dX Complex symplectic Fourier transform : F symp f ( ξ ) = F C N f ( J ξ ) F symp f ( ξ ) = Z C N f ( X ) e − iπ Re ω ( X,ξ ) dX Partial Fourier transform : on C N ≃ C n × C n F C n f ( X , ξ ) = Z C n f ( X , X ) e − iπ Re h X ,ξ i dX Real conjugate Fourier transform : F ε f ( ξ ) = F R N f ( εξ ) F ε f ( ξ , ξ ) = F C N f ( ξ + iξ ) F ε f ( ξ ) = Z R N × R N f ( X ) e − iπ ( h X ,ξ i−h X ,ξ i ) d ( X , X ) Restriction to
Sp( n ) The determination of a K -type formula for π µ,δ relies on some knownfacts regarding the representation theory of orthogonal groups over R , C and H . More precisely it will involve the isotypical decompositions of square-integrable functions over the Euclidean unit sphere in R n ≃ C n ≃ H n .3.1. Real, complex and quaternionic spherical harmonics.
Let us fixthe following classical identifications:(4) H n ≃ ( C n + j C n ) ≃ (( R n + i R n ) + j ( R n + i R n )) where i , j and k = ij denote the standard quaternion units. Then the unitspheres S ( · ) of those isometric vector spaces all identify to S N − and carrycompatible left actions of the corresponding orthogonal groups as follows:(5) Sp( n ) y S ( H n ) x Sp(1) ≃ SU(2) ∩ ≃ ∪ U(2 n ) y S ( C n ) x U(1) ∩ ≃ ∪ O(4 n ) y S ( R n ) x {± } ≃ S N − The right column displays the right actions of scalars of norm . As anidentification between Sp(1) and
SU(2) , we fix the one given by(6) i (cid:20) i − i (cid:21) , j (cid:20) − (cid:21) , k (cid:20) ii (cid:21) , so that U(1) naturally appears as a Cartan subgroup of
Sp(1) via the map(7) e iθ (cid:20) e iθ e − iθ (cid:21) . Square integrable functions on S N − decompose with respect to the cha-racters of {± } as even and odd, while the component corresponding to δ ∈ Z ≃ [ U(1) is the space L (cid:0) S N − (cid:1) δ introduced in Section 1.3.2. Noambiguity arises from the fact that the action was defined on the left there,since U(1) is abelian.Let us now turn to the decomposition of L (cid:0) S N − (cid:1) into irreducible rep-resentations of O(4 n ) and SU(2 n ) , that is the classical theory of sphericalharmonics on real and complex vector spaces. More details may be found in[2, Section 2.1].As in Section 2 we denote by H k ( R N ) the vector space of harmonic ho-mogeneous polynomials on R N of degree k ∈ N . It is also useful to considerthe space H α,β ( C N ) of harmonic polynomials of the complex variable andits conjugate, homogeneous of degree α in Z ∈ C N and of degree β in ¯ Z .Under the identifications of (4), there is a natural isomorphism H k ( R N ) ≃ M α + β = k H α,β ( C N ) . Restricting functions to the sphere provides a complete orthogonal basis,hence a discrete sum decomposition of L (cid:0) S N − (cid:1) into irreducible compo-nents of the left actions of O(4 n ) and U(2 n ) in (5), namely(8) L (cid:0) S N − (cid:1) ≃ X ⊕ k ≥ H k ( R N ) (cid:12)(cid:12)(cid:12) S N − ≃ X ⊕ k ≥ M α + β = k H α,β ( C N ) (cid:12)(cid:12)(cid:12) S N − . Taking into account the right actions in (5), one can refine (8) as L (cid:0) S N − (cid:1) even ≃ X ⊕ k ∈ N H k ( R N ) (cid:12)(cid:12)(cid:12) S N − (9) L (cid:0) S N − (cid:1) odd ≃ X ⊕ k ∈ N +1 H k ( R N ) (cid:12)(cid:12)(cid:12) S N − (10)in the real case and(11) L (cid:0) S N − (cid:1) δ ≃ X ⊕ β − α = δ H α,β ( C N ) (cid:12)(cid:12)(cid:12) S N − in the complex case. EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) Remark 4.
The above discussion of the isotypical decomposition of L (cid:0) S N − (cid:1) with respect to the action of U(2 n ) × U(1) (resp.
O(4 n ) × Z ) only involvesthe irreducible representations of U(2 n ) (resp. O(4 n ) ) because the commu-tativity of U(1) (resp. Z ) implies that these groups have 1-dimensionalirreducible unitary representations, which is not the case of Sp(1) .In order to proceed with the same analysis over quaternions and writedown the analogue of (8) corresponding to the action of
Sp( n ) × Sp(1) on L (cid:0) S N − (cid:1) , some additional notations are needed. Following [5, Sections 5and 6], we denote by V l,l ′ n the unique unitary irreducible representation of Sp( n ) corresponding to the highest weight ( l, l ′ , , . . . , where l and l ′ areintegers satisfying l ≥ l ′ ≥ . Similarly, V j denotes the irreducible j + 1 -dimensional representation of Sp(1) ≃ SU(2) .Since
U(1) naturally embeds into
SU(2) via (7), this representation de-composes according to the characters of the circle. More precisely, if C δ denotes the space of the character z z δ of U(1) , for δ ∈ Z , then(12) V j ≃ M | δ |≤ jδ ≡ j [2] C δ . Now identifying S N − to S ( H n ) , the isotypic decomposition of L (S N − ) with respect to action of Sp( n ) × Sp(1) defined by the first line of (5) is givenin [5] by: L (S N − ) ≃ X ⊕ l ≥ l ′ ≥ V l,l ′ n ⊗ V l − l ′ . Together with (12), this decomposition gives(13) L (S N − ) ≃ X ⊕ ( δ, ( l,l ′ )) ∈ Z × N , (cid:26) l − l ′ ≥| δ | l − l ′ ≡ δ [2] V l,l ′ n ⊗ C δ under the action of Sp( n ) × U(1) .3.2.
The branching law
Sp( n, C ) ↓ Sp( n ) . The above discussion leads tothe determination of the branching law
Sp( n, C ) ↓ Sp( n ) of π µ,δ seen in thecompact picture. Proposition 2 ( K -type formula) . The restriction of π µ,δ to the maximalcompact subgroup Sp( n ) decomposes into irreducible components as follows: π µ,δ | Sp( n ) ≃ X ⊕ l − l ′ ≥| δ | l − l ′ ≡ δ [2] V l,l ′ n . Every K -type V l,l ′ n occurs at most once in this decomposition.Proof. The discussion of Paragraph 1.3.2 shows how π µ,δ can be realised on L (cid:0) S N − (cid:1) δ . In this picture, the restriction to K of the action coincides with the natural representation of Sp( n ) on functions over the unit sphere of H n . Subsequently, the first statement reduces to fixing δ in (13). The factthat this decomposition is multiplicity-free relies on the observation thatalthough a summand C δ appears in V l − l ′ for various values of l − l ′ , thesedo not involve the same space V l,l ′ n more than once. (cid:3) Action of the Knapp-Stein operators on the K -types. The lastresult of this section describes the behaviour of the Knapp-Stein intertwinerson each K -type. Proposition 1 proves that e T µ,δ intertwines π µ,δ and π − µ, − δ .As a consequence of the formula in Proposition 2, these representations havethe same K -types V l,l ′ n . Taking into account the right action of U(1) seen aCartan subgroup of
Sp(1) via (5) and the corresponding isotypic decompo-sition (13) of L (cid:0) S N − (cid:1) , one is led to study the restriction T l,l ′ µ,δ : V l,l ′ n ⊗ C δ −→ V l,l ′ n ⊗ C − δ ∩ ∩ V l,l ′ n ⊗ V l − l ′ V l,l ′ n ⊗ V l − l ′ . In order to specifiy how T µ,δ restricts to an operator of V l,l ′ n , it is neces-sary to fix an identification between V l,l ′ n ⊗ C δ and V l,l ′ n ⊗ C − δ . Using theisomorphism (6) between Sp(1) and
SU(2) , it is done by choosing a non-trivial element w in the Weyl group W (SU(2) : U(1)) and letting it act byconjugation on SU(2) . Since such an action inverts the elements in the torus
U(1) , it provides an isomorphism ι w from V l,l ′ n to itself, that exchanges thesummands C δ and C − δ appearing in (12):(14) ι w : V l − l ′ ∼ −→ V l − l ′ ∪ ∪ C δ ∼ −→ C − δ . The restricted intertwining operator under this identification will be de-noted by T l,l ′ ,wµ,δ : T l,l ′ ,wµ,δ = T l,l ′ µ,δ ⊗ ι w . From now on, w will be the class modulo U(1) of (cid:20) − (cid:21) . Since thismatrix of SU(2) normalises
U(1) , it defines a Weyl element, hence fixes theabove definition of T l,l ′ ,wµ,δ . Proposition 3.
Let δ = 0 . For l, l ′ ∈ N such that l − l ′ ≥ | δ | and l − l ′ ≡ δ [2] ,the restriction T l,l ′ ,wµ,δ of the normalised Knapp-Stein intertwiner e T µ,δ acts on V l,l ′ n as the scalar π − µ ( − i ) − ( l + l ′ ) Γ (cid:16) n + l + l ′ + µ (cid:17) Γ (cid:16) n + l + l ′ − µ (cid:17) . EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) Proof.
Let p ∈ V l,l ′ n ⊗ C δ ⊂ V l,l ′ n ⊗ V l − l ′ . By compatibility of the isotypicdecompositions (8) and (13), p can be seen as the restriction to S N − of apolynomial in H l + l ′ ( R N ) . In view of Proposition 1 and Lemma 3, the oper-ator e T µ,δ maps p µ − N to B N ( µ − N, l + l ′ ) p − µ − N ( J ε · ) . Under the identifica-tions (4), applying J ε to vectors in H n from the left amounts to multiplyingthem by the quaternionic unit j from the right. It follows that T l,l ′ ,wµ,δ p = B N ( µ − N, l + l ′ ) ι w ( p.j ) where j ∈ Sp(1) acts via V l − l ′ . Identifying Sp(1) to SU(2) by (6) again,we realise V l − l ′ as the classical representation of SU(2) on homogeneouspolynomials of degree l − l ′ in two variables x and y , denoted by Φ l − l ′ [ x, y ] .Since j and w are both represented by the matrix (cid:20) − (cid:21) , one has ι w ( p.j ) = p. ( − I ) = ( − δ p, hence the conclusion: T l,l ′ ,wµ,δ p = ( − δ B N ( µ − N, l + l ′ ) p . (cid:3) Analysis of π , . Let us discuss the elements of the dual space b G obtained from the degenerate principal series, that is the equivalence classesof irreducible unitary subrepresentations of { π iλ,δ , R × Z } . It is proved in [4]that π iλ,δ is irreducible if ( λ, δ ) = (0 , . Moreover, the Knapp-Stein operator e T iλ,δ (or its algebraic version T iλ,δ in the non-standard model) exhibits aunitary equivalence between π iλ,δ and π − iλ, − δ . It is also established in [4]that π , splits into the direct sum of two irreducible subrepresentations.The following result describes this splitting in terms of eigenspaces of theKnapp-Stein operators and specifies the K -module structure of the sum-mands. Theorem 1.
The representation π , of G is reducible and decomposes as π , ≃ π − , ⊕ π +0 , , where the irreducible summands π ± , are characterised by:(1) their K -type formula: π − , ≃ X ⊕ l − l ′ ≡ V l,l ′ n and π +0 , ≃ X ⊕ l − l ′ ≡ V l,l ′ n ; (2) the classical Knapp-Stein intertwiners: π +0 , (resp. π − , ) is the eigenspacefor the eigenvalue (resp. − ) of e T , acting on V , .(3) the algebraic Knapp-Stein intertwiners: π +0 , (resp. π − , ) is the eigenspacefor the eigenvalue (resp. − ) of T , acting on L (cid:0) C m +1 (cid:1) .Proof. To establish (1) and (2) , we proceed as in the proof of Proposition3, except that no choice of a Weyl representative is required to identify V l − l ′ ⊗ C δ to V l − l ′ ⊗ C − δ when δ = 0 . For p ∈ V l,l ′ n ⊗ C ⊂ V l,l ′ n ⊗ V l − l ′ one has e T , p = B N ( − N, l + l ′ )( p.j ) where j acts by the representation V l − l ′ of Sp(1) . Identifying
Sp(1) to SU(2) so that j is represented by (cid:20) − (cid:21) , and V l − l ′ to Φ l − l ′ [ x, y ] , it appears that j acts on the variables by ( x, y ) ( y, − x ) so that a 0-weight vector ξ is send to ( − l − l ′ ξ by j . It follows that theintertwining operator acts on every K -type V l,l ′ n of π , by B N ( − N, l + l ′ )( − l − l ′ . Since B N ( − N, l + l ′ ) = ( − − l + l ′ by definition, it follows that e T , p = ( − − l ′ p, hence the result. We postpone the proof of (3) to the next paragraph wherethe algebraic Knapp-Stein operators are defined and studied. (cid:3) uuu uu uqqq qq q u : V l,l ′ n appears in π +0 , q : V l,l ′ n appears in π − , l l ′ Figure 1.
Repartition of the K -types of π , A non-standard model and intertwining operators
This section is devoted to the description of a new model of degenerateprincipal series, in which intertwining operators happen to take an algebraicform.4.1.
Non-standard model.
Let µ and δ be as above. The non-compactpicture described in Paragraph 1.3.3 allowed to realise π µ,δ on the Hilbertspace L (cid:0) H m +1 C (cid:1) ≃ L ( C × C m × C m ) ≃ L (cid:0) C m +1 (cid:1) . EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) Let F C × C m be the partial Fourier transform defined on L ( C × C m × C m ) by F C × C m f ( τ, X , ξ ) = Z C × C m f ( t, X , X ) e − iπ Re ( tτ + h X ,ξ i ) dX dt. Definition 4.
The non-standard model U µ,δ of π µ,δ is the image of the non-compact picture L (H m +1 C ) by F C × C m : L (cid:0) C m +1 (cid:1) −→ L (cid:0) C m +1 (cid:1) , that is U µ,δ = L (cid:0) C m +1 (cid:1) as a Hilbert space and the action of an element g ∈ G on U µ,δ is given by F C × C m ◦ π µ,δ ( g ) ◦ F − C × C m .The equivalences between the induced, non-compact and non-standardmodels of π µ,δ are summed up in the following diagram:(15) α µ,δ : V µ,δ ∼ restrict . / / L (H m +1 C ) ∼F C × C m / / U µ,δ f ✤ / / F ✤ / / H where, according to the embedding (1), F ( t, X , X ) = f (1 , X , t, X ) for t ∈ C and ( X , X ) ∈ C m × C m . Lemma 4.
Let f ∈ V µ,δ . With notations as above, H ( τ, X , ξ ) = 12 F C n f (cid:16) , X , τ , ξ (cid:17) . Proof.
According to the notations in the definition (15) of α µ,δ , one has H ( τ, X , ξ ) = Z C × C m F ( t, X , X ) e − iπ Re ( tτ + h X ,ξ i ) dX dt = Z C n f (1 , X , t, X ) e − iπ Re h ( t,X ) , ( τ,ξ ) i d ( t, X )= 12 Z C n f (1 , X , t, X ) e − iπ Re h ( t,X ) , ( τ ,ξ ) i d ( t, X ) , hence the result. (cid:3) Algebraic Knapp-Stein intertwiners.
We now come to the mainpoint of this section, id est the proof that the normalised Knapp-Stein op-erators considered in Section 2 take an algebraic form once expressed inthe non-standard model of the previous paragraph. More precisely, for H ∈ L (cid:0) C m +1 (cid:1) , we let T µ,δ H ( s, X , X ) = (cid:12)(cid:12)(cid:12) s (cid:12)(cid:12)(cid:12) − µ [ s ] − δ H (cid:18) s, s X , s X (cid:19) . The jacobian determinant of the transform ( s, X , X ) (cid:18) s, s X , s X (cid:19) is easily seen to have modulus , so that T µ,δ is an endomorphism of L (cid:0) C m +1 (cid:1) ,which turns out to be the realisation of the normalised Knapp-Stein inter-twiner e T µ,δ in the non-standard picture: Theorem 2.
For any ( µ, δ ) ∈ i R × Z , the following diagram is commutative: V − µ, − δ e T µ,δ / / α − µ, − δ (cid:15) (cid:15) V µ,δα µ,δ (cid:15) (cid:15) U − µ, − δ T µ,δ / / U µ,δ . Proof.
Let f ∈ V ∞− µ, − δ . Then α µ,δ ◦ e T µ,δ f ( τ, X , ξ ) = F C × C m F symp f ( τ, X , ξ ) by Proposition 1, = 12 F C n F symp f (cid:16) , X , τ , ξ (cid:17) by Lemma 4, = 12 F C n f (cid:16) τ , ξ , , X (cid:17) by Lemma 2, = 12 (cid:12)(cid:12)(cid:12)(cid:12) τ (cid:12)(cid:12)(cid:12)(cid:12) µ (cid:20) τ (cid:21) δ F C n f (cid:18) , τ ξ , τ , τ X (cid:19) by Lemma 1, = (cid:12)(cid:12)(cid:12) τ (cid:12)(cid:12)(cid:12) − µ [ τ ] − δ F C n f (cid:18) , τ ξ , τ , τ X (cid:19) = (cid:12)(cid:12)(cid:12) τ (cid:12)(cid:12)(cid:12) − µ [ τ ] − δ H (cid:18) τ, τ ξ , τ X (cid:19) by Lemma 4,thus proving that α µ,δ ◦ e T µ,δ = T µ,δ ◦ α − µ, − δ (cid:3) We can now complete the proof of Theorem 1. Since T , = Id L ( C m +1 ) ,every function H ∈ L (cid:0) C m +1 (cid:1) can be written in a unique way as H = H + + H − with T , H + = H + and T , H − = − H − . Indeed, H ± = 12 ( H ± T , H ) , which gives the expected characterisation (3) of π ± , in Theorem 1.4.3. Perspectives.
The existence of a non-standard model for degenerateprincipal series of
Sp( n, R ) and Sp( n, C ) in which the Knapp-Stein intertwin-ers take an algebraic form relies on the special form of the nilradical of theinducing parabolic subgroup. It is then natural to ask if the same occurswith parabolic subgroups of Heisenberg type in other groups. EGENERATE PRINCIPAL SERIES OF
Sp( n, C ) Acknowledgements
The results presented here were mostly obtained during the author’s stay atthe Gradutate School of Mathematical Sciences of the University of Tokyo.We wish to heartily thank Pr. Toshiyuki Kobayashi for his kindness as a hostand many enlightening discussions during the preparation of this article. Wealso thank Pr. Pevzner for helpful discussions and the meetings he organisedin Reims.
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Pierre Clare, The Pennsylvania State University, Department of Mathe-matics, McAllister Building, University Park, PA - 16802
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