Transfer of characters for discrete series representations of the unitary groups in the equal rank case via the Cauchy-Harish-Chandra integral
aa r X i v : . [ m a t h . R T ] J a n TRANSFER OF CHARACTERS FOR DISCRETE SERIES REPRESENTATIONS OF THE UNITARYGROUPS IN THE EQUAL RANK CASE VIA THE CAUCHY-HARISH-CHANDRA INTEGRAL
ALLAN MERINOA bstract . As conjectured by T. Przebinda, the transfer of characters in the Howe’s correspondence should be ob-tained via the Cauchy-Harish-Chandra integral. In this paper, we prove that the conjecture holds for the dual pair(G = U( p , q ) , G ′ = U( r , s )), p + q = r + s , starting with a discrete series representation Π of e U( p , q ). C ontents
1. Introduction 12. Howe correspondence and Cauchy-Harish-Chandra integral 33. Explicit formulas of Chc for unitary groups 54. Transfer of invariant eigendistributions 115. Discrete series representations and a result of A. Paul 136. Proof of Conjecture 4.7 for discrete series representations in the equal rank case 157. A commutative diagram and a remark on the distribution Θ Π ′ ntroduction Let W be a finite dimensional vector space over R endowed with a non-degenerate, skew-symmetric, bilinearform h· , ·i , Sp(W) be the corresponding group of isometries, f Sp(W) be the metaplectic cover of Sp(W) (see [1,Definition 4.18]) and ( ω, H ) be the corresponding Weil representation (see [1, Section 4.8]). For every irreduciblereductive dual pair (G , G ′ ) in Sp(W), R. Howe proved (see [14, Theorem 1]) that there is a bijection between R ( e G , ω ) and R ( f G ′ , ω ) whose graph is R ( e G · f G ′ , ω ) (where R ( e G , ω ) is defined in Section 2). More precisely, toevery Π ∈ R ( e G , ω ), we associate a representation finitely generated admissible representation Π ′ of f G ′ which hasa unique irreducible quotient Π ′ such that Π ⊗ Π ′ ∈ R ( e G · f G ′ , ω ). We denote by θ : R ( e G , ω ) ∋ Π → Π ′ = θ ( Π ) ∈ R ( f G ′ , ω ) the corresponding bijection.As proved by Harish-Chandra (see [7, Section 5] or Section 4), all the representations Π of e G (resp. Π ′ of f G ′ )appearing in the correspondence admit a character, i.e. a e G-invariant distribution Θ Π on e G (in the sense of LaurentSchwartz) given by a locally integrable function Θ Π on e G which is analytic on e G reg (the set of regular elements of Mathematics Subject Classification.
Primary: 22E45; Secondary: 22E46, 22E30.
Key words and phrases.
Howe correspondence, Characters, Cauchy–Harish-Chandra integral, Orbital Integrals, Discrete SeriesRepresentations. G). The character Θ Π determines the representation Π . In particular, one way to understand the Howe correspon-dence, i.e. to make the map θ explicit, is to understand the transfer of characters.In his paper [23], T. Przebinda conjectured that the correspondence of characters should be obtained via the so-called Cauchy-Harish-Chandra integral that he introduced in [23]. We recall briefly the construction of this integral.Let T : f Sp(W) → S ∗ (W) be the embedding of the metaplectic group inside the space of tempered distributionson W as in [1, Definition 4.23] (see also Remark 2.2) and H , . . . , H n be a maximal set of non-conjugate Cartansubgroups of G which are ι -invariant, where ι is a Cartan involution on G. Every Cartan subgroup H i can bedecomposed as H i = T i A i , with T i maximal compact in H i . Let A ′ i and A ′′ i be the subgroups of Sp(W) defined byA ′ i = C Sp(W) (A i ) and A ′′ i = C Sp(W) (A ′ i ). One can easily check that (A ′ i , A ′′ i ) form a dual pair in Sp(W), which is notirreducible in general. For every function ϕ ∈ C ∞ c ( f A ′ i ), we define Chc( ϕ ) byChc( ϕ ) = Z A ′′ i \ W A ′′ i T( ϕ )( w ) dw , where dw is a measure on the manifold A ′′ i \ W A ′′ i defined in [23, Section 1]. As mentioned in [23, Section 2] (seealso Section 2), Chc( Ψ ) is well-defined and the corresponding map Chc : C ∞ c ( f A ′ i ) → C is a distribution on f A ′ i . Forevery regular element ˜ h i ∈ e H i reg , we denote by Chc ˜ h i the pull-back of Chc through the map f G ′ ∋ ˜ g ′ → ˜ h i ˜ g ′ ∈ f A ′ i .Assume now that rk(G) ≤ rk(G ′ ). In [3] (see also Section 4), F. Bernon and T. Przebinda defined a map:Chc ∗ : D ′ ( e G) e G → D ( f G ′ ) f G ′ , where D ′ ( e G) e G is the set of e G-invariant distributions on e G. More precisely, if Θ is a e G-invariant distribution givenby a locally integrable function Θ on e G, then, for every ϕ ∈ C ∞ c ( f G ′ ), we get:Chc ∗ ( Θ )( ϕ ) = n X i = | W (H i ) | Z e H i reg Θ (˜ h i ) | det(1 − Ad(˜ h − i )) g / h i | Chc ˜ h i ( ϕ ) d ˜ h i . The conjecture can be stated as follows:
Conjecture 1.1.
Let G and G ′ be the Zariski identity components of G and G ′ respectively. Let Π ∈ R ( e G , ω )satisfying Θ Π | e G / f G1 = = O(V), where V is an even dimensional vector space over R or C . Then, up to aconstant, Chc ∗ ( Θ Π ) = Θ Π ′ on f G ′ .This result is well-known if the group G is compact and had been proved recently in [24] in the stable range. Inthis paper, we investigate the case (G , G ′ ) = (U( p , q ) , U( r , s )), p + q = r + s ( p will always be assumed to be smalleror equal than q , in particular, the number of non-conjugate Cartan subgroups of G is p + Π ∈ R ( e G , ω ) a discrete series representation of e G. Let λ be the Harish-Chandra parameter of Π . In this case, usingLi’s result (see [17, Proposition 2.4] or Section 7), we get that Π ′ = Π ′ and using [20], we know that Π ′ = θ ( Π )is a discrete series representations of f G ′ (with Harish-Chandra parameter λ ′ ), and the correspondence λ → λ ′ isknown and explicit (see [20, Theorem 2.7]).In order to prove that, up to a constant, Chc ∗ ( Θ Π ) = Θ Π ′ = Θ Π ′ , we use a parametrisation of discrete seriescharacters provided by Harish-Chandra (see [8, Lemma 44]). More precisely, it follows from [3] and a result ofHarish-Chandra (see [9, Theorem 2]) that the distribution Chc ∗ ( Θ Π ) is given by locally integrable function Θ ′ Π analytic on f G ′ reg . Using [3, Theorem 2.2], we proved in Proposition 6.5 that the value of Θ ′ Π on f H ′ reg , where H ′ isthe compact Cartan subgroup of G ′ , is of the form: ∆ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = C X σ ∈ S r × S s ε ( σ )( σ ˇ h ′ ) λ Π , (ˇ h ′ ∈ ˇH ′ reg ) , here C ∈ R , ˇH ′ is a double cover of H ′ (see Section 3) chosen such that ρ ′ = P α> α is analytic integral, ˇ p isa map from ˇH ′ into f H ′ (which is not an isomorphism of double covers in general), and λ Π is a linear form on i h ′ depending on Π which is conjugated to λ under S r + s . Moreover, using results of [2], we proved in Proposition 6.7that sup ˜ g ′ ∈ f G ′ reg | D( ˜ g ′ ) | | Θ ′ Π ( ˜ g ′ ) | < ∞ , where D is the Weyl denominator defined in Notations 5.3. Finally, applying [3, Theorem 1.3] to our particulardual pair, it follows that z Chc ∗ ( Θ Π ) = χ λ Π ( z )Chc ∗ ( Θ Π ) for every z ∈ Z( U ( g ′ C )), where χ λ Π is the character ofZ( U ( g ′ C )) obtained via the linear form λ Π as in Remark A.9 and then, using a result of Harish-Chandra (see [8,Lemma 44]) we get that Chc ∗ ( Θ Π ) is the character of a discrete series representations of f G ′ with Harish-Chandraparameter λ Π .In Section 7, we prove, by using results of [22] (see also [17]), that T( Θ Π ) is a well-defined e G · f G ′ -invariantdistribution on S ∗ (W) and we get in Corollary 7.4 the following equality:T( Θ Π ) = C Π ⊗ Π ′ T(Chc ∗ ( Θ Π )) , where C Π ⊗ Π ′ is a constant depending on Π and Π ′ . In particular, we can hope that the following diagram oftencommutes (up to a constant): D ′ ( e G) e G Chc ∗ / / T % % ❏❏❏❏❏❏❏❏❏ D ′ ( f G ′ ) f G ′ T y y ssssssssss S ∗ (W) e G · f G ′ Moreover, according to Li’s result (see [17] or Section 7), Π can be embedded in ω as a subrepresentation, and byprojecting onto the ν ⊗ Π ′ -isotypic component (where ν is the lowest e K-type of Π as in Theorem 5.4), we get (seeEquation (13)) the following equality:Chc ∗ ( Θ Π )( ϕ ) = d Π tr Z e K Z e G Z f G ′ Θ Π ν (˜ k ) Θ Π ( ˜ g ) ϕ ( ˜ g ′ ) ω (˜ k ˜ g ˜ g ′ ) d ˜ g ′ d ˜ gd ˜ k ! , where ϕ ∈ C ∞ c ( f G ′ ) and d Π is the formal degree of Π (see Remark 5.2). Acknowledgements:
The motivation of this paper comes from a talk given by Wee Teck Gan at the RepresentationTheory and Number theory seminar at NUS in 2019. I would like to thank Tomasz Przebinda for the many usefuldiscussions during the preparation of this paper. This research was supported by the MOE-NUS AcRF Tier 1grants R-146-000-261-114 and R-146-000-302-114.2. H owe correspondence and C auchy -H arish -C handra integral Let W be a finite dimensional vector space over R endowed with a non-degenerate, skew-symmetric, bilinearform h· , ·i . We denote by Sp(W) the corresponding group of isometries, i.e.Sp(W) = (cid:8) g ∈ GL(W) , h g ( w ) , g ( w ′ ) i = h w , w ′ i , ( ∀ w , w ′ ∈ W) (cid:9) , and by f Sp(W) the metaplectic group as in [1, Definition 4.18]: it’s a connected two-fold cover of Sp(W). We willdenote by π : f Sp(W) → Sp(W) the corresponding covering map. e say that a pair of subgroup (G , G ′ ) of Sp(W) is a dual pair if G is the centralizer of G ′ in Sp(W) and vice-versa.The dual pair is said to be reductive if both G and G ′ act reductively on W and irreducible if we cannot find anorthogonal decomposition of W = W ⊕ W where both W and W are G · G ′ -invariant. One can easily prove thatthe preimages e G = π − (G) and f G ′ = π − (G ′ ) in f Sp(W) form a dual pair in f Sp(W).Let ( ω, H ) be the Weil representation of f Sp(W) corresponding to a fixed unitary character of R and ( ω ∞ , H ∞ ) bethe corresponding smooth representation (see [1, Section 4.8]). For a subgroup e H of f Sp(W), we denote by R ( e H , ω )the set of conjugacy classes of irreducible admissible representations ( Π , H Π ) of e H which can be realized as aquotient of H ∞ by a closed ω ∞ ( e H)-invariant subspace.As proved by R. Howe (see [14, Theorem 1]), for every reductive dual pair (G , G ′ ) of Sp(W), we have a one-to-onecorrespondence between R ( e G , ω ) and R ( e G ′ , ω ) whose graph is R ( e G · f G ′ , ω ). More precisely, if Π ∈ R ( e G , ω ),we denote by N( Π ) the intersection of all the closed e G-invariant subspaces N such that Π ≈ H ∞ / N . Then, thespace H ( Π ) = H ∞ / N( Π ) is a e G · f G ′ -module; more precisely, H ( Π ) = Π ⊗ Π ′ , where Π ′ is a f G ′ -module, notirreducible in general, but Howe’s duality theorem says that there exists a unique irreducible quotient Π ′ of Π ′ with Π ′ ∈ R ( f G ′ , ω ) and Π ⊗ Π ′ ∈ R ( e G · f G ′ , ω ).We will denote by θ : R ( e G , ω ) → R ( f G ′ , ω ) the corresponding bijection. Notation 2.1.
We use here the notations of [1]. We denote by S ∗ (W) the space of tempered distributions on W andby T : f Sp(W) → S ∗ (W)the injection of f Sp(W) into S ∗ (W) (see [1, Definition 4.23]). We denote by Sp c (W) the subset of Sp(W) given by (cid:8) g ∈ Sp(W) , det( g − , (cid:9) and by f Sp c (W) its preimage in f Sp(W).
Remark . As explained in [1], for every ˜ g ∈ f Sp c (W), the distribution T( ˜ g ) is defined by T( ˜ g ) = Θ ( ˜ g ) χ c ( g ) µ W ,where Θ is the character of the Weil representation ( ω, H ) defined in [1, Definition 4.23], χ c ( g ) : W → C isthe function on W given by χ c ( g ) ( w ) = χ (cid:16) h ( g + g − − w , w i (cid:17) with g = π ( ˜ g ) and µ W is the appropriatelynormalized Lebesgue measure on W.The map T can be extended to ] Sp(W) and to C ∞ c ( f Sp(W)) byT( ϕ ) = Z f Sp(W) ϕ ( ˜ g )T( ˜ g ) d ˜ g , ( ϕ ∈ C ∞ c ( f Sp(W))) , where d ˜ g is the Haar measure on f Sp(W). As proved in [1, Section 4.8], for every ϕ ∈ C ∞ c ( f Sp(W)), the distributionT( ϕ ) on W is given by a Schwartz function on W still denoted by T( ϕ ), i.e.T( ϕ )( φ ) = Z W T( ϕ )( w ) φ ( w ) d µ W ( w ) , ( φ ∈ S(W)) . We now recall the construction of the Cauchy-Harish-Chandra integral introduced by T. Przebinda in [23].We denote by H i , 1 ≤ i ≤ n be a maximal set of non-conjugate Cartan subgroups of G. As explained in [26,Section 2.3.6], for every 1 ≤ i ≤ n , the Cartan subgroup H i can be decomposed as H i = T i A i , with T i maximalcompact in H i (A i is called the split part of H i ). For 1 ≤ i ≤ n , we denote by A ′ i and A ′′ i the subgroups of Sp(W)given by A ′ i = C Sp(W) (A i ) and A ′′ i = C Sp(W) (A ′ i ). As recalled in [23, Section 1], there exists an open and dense ubset W A ′′ i , which is A ′′ i -invariant and such that A ′′ i \ W A ′′ i is a manifold, endowed with a measure dw such thatfor every φ ∈ C ∞ c (W) such that supp( φ ) ⊆ W A ′′ i , Z W A ′′ i φ ( w ) d µ W ( w ) = Z A ′′ i \ W A ′′ i Z A ′′ i φ ( aw ) dadw . For every ϕ ∈ C ∞ c ( e A ′ i ), we denote by Chc( ϕ ) the following integral:Chc( ϕ ) = Z A ′′ i \ W A ′′ i T( ϕ )( w ) dw . According to Remark 2.2, the previous integral is well-defined and as proved in [23, Lemma 2.9], the correspondingmap Chc : C ∞ c ( e A ′ i ) → C defines a distribution on e A ′ i .For every ˜ h i ∈ e H i , we denote by τ ˜ h i the map: τ ˜ h i : f G ′ ∋ ˜ g ′ → ˜ h ˜ g ′ ∈ e A ′ i and, for ˜ h i regular, by Chc ˜ h i = τ ∗ ˜ h i (Chc), where τ ∗ ˜ h i is the pull-back of τ ˜ h i as defined in [13, Theorem 8.2.4]. Inparticular, for every ˜ h i ∈ e H i reg , Chc ˜ h i is a well-defined distribution on f G ′ .3. E xplicit formulas of Chc for unitary groups
Let V = C p + q and V ′ = C r + s be two complex vector spaces endowed with non-degenerate bilinear forms ( · , · )and ( · , · ) ′ respectively, with ( · , · ) hermitian and ( · , · ) ′ skew-hermitian, and let ( p , q ) (resp. ( r , s )) be the signature of( · , · ) (resp. ( · , · ) ′ ). We assume that p + q ≤ r + s . Let B V = { e , . . . , e n } , n = p + q (resp. B V ′ = n e ′ , . . . , e ′ n ′ o , n ′ = r + s ) be a basis of V (resp. V ′ ) such that Mat(( · , · ) , B V ) = Id p , q (resp. Mat(( · , · ) ′ , B V ′ ) = i Id r , s ). Let G andG ′ be the corresponding group of isometries, i.e.G = G(V , ( · , · )) ≈ n g ∈ GL( n , C ) , g t Id p , q g = Id p , q o , G ′ = G(V ′ , ( · , · ) ′ ) ≈ n g ∈ GL( n ′ , C ) , g t Id r , s g = Id r , s o . where ≈ is a Lie group isomorphism.Let H and H ′ be the diagonal compact Cartan subgroups of G and G ′ respectively. By looking at the action of Hon the space V, we get a decomposition of V of the form:V = V ⊕ . . . ⊕ V n , where the spaces V a given by V a = C ie a are irreducible H-modules. We denote by J the element of h such thatJ = i Id V and let J j = i E j , j . Similarly, we write V ′ = V ′ ⊕ . . . ⊕ V ′ n ′ , with V ′ b = C ie ′ b , J ′ the element of h ′ given by J ′ = i Id V ′ and J ′ j = i E j , j . Let W = Hom C (V ′ , V) endowed with thesymplectic form h· , ·i given by: h w , w i = tr C / R ( w ∗ w ) , ( w , w ∈ W) , where w ∗ is the element of Hom(V , V ′ ) satisfying: (cid:0) w ∗ ( v ′ ) , v (cid:1) = (cid:0) v ′ , w ( v ) (cid:1) ′ ( v ∈ V , v ′ ∈ V ′ ) . The space W can be seen as a complex vector space by(1) i w = J ◦ w ( w ∈ W) . e define a double cover g GL C (W) of the complex group GL C (W) by: g GL C (W) = n ˜ g = ( g , ξ ) ∈ GL C (W) × C × , ξ = det( g ) o . Because p + q ≤ r + s , we get a natural embedding of h C into h ′ C and we denote by Z ′ = G ′ h the centralizer of h inG ′ . Let H ′ C be the complexification of H ′ in GL C (W). In particular, H ′ C is isomorphic to h ′ C / n ′ X j = π x j J j , x j ∈ Z . We denote by ˇH ′ C the connected two-fold cover of H C isomorphic to h ′ C / n ′ X j = π x j J ′ j , n ′ X j = x j ∈ Z , x j ∈ Z . Let p : ˇH ′ C → H C the corresponding covering map. If ˇH ′ C is isomorphic to e H ′ C , we may choose an isomorphismˇ p : ˇH ′ C → e H ′ C so that p = ˜ p ◦ ˇ p . Otherwise, e H ′ C coincides with the direct product H C , × {± } . In this case, we candefine ˇ p : ˇH ′ C → ˜H C to be the composition of p with the inclusion H C → H C × {± } . Then, again p = ˜ p ◦ ˇ p . Remark . (1) Let Ψ ′ : = Ψ ′ ( g ′ C , h ′ C ) be a set of positive roots corresponding to ( g ′ C , h ′ C ), Ψ ′ ( k ′ ) be a the set ofcompact roots in Ψ ′ , where k is the Lie algebra of K = U( r ) × U( s ), and Ψ ′ n be the set of non-compact rootsof Ψ ′ , i.e. Ψ ′ n = Ψ ′ \ Ψ ′ ( k ). The reason why we are considering the double cover ˇH ′ C of H ′ C is to make theform ρ ′ = P α ∈ Ψ ′ α analytic integral. For every analytic integral form γ on h ′ C , we will denote by ˇ h ′ → ˇ h ′ γ the corresponding character on H ′ C .(2) We know that, up to conjugation, the number of Cartan subgroups in U( r , s ) is min( r , s ) +
1. Those Cartansubgroups can be parametrized by some particular subsets of Ψ ′ n . Let Ψ ′ st n be the set of strongly orthogonalroots in Ψ ′ (see [25, Section 2]).For every α ∈ Ψ ′ st n , we denote by c ( α ) the element of G ′ C given by: c ( α ) = exp (cid:18) π X − α − X α ) (cid:19) . where X α (resp. X − α ) is in g ′ C ,α (resp. g ′ C , − α ) and normalized as in [25, Equation 2.7]. For every subset Sof Ψ ′ st n , we denote by c (S) the following element of G ′ C defined by c (S) = Y α ∈ S c ( α ) , and let h ′ (S) = g ′ ∩ Ad( c (S))( h ′ C ) . We denote by H ′ (S) the analytic subgroup of G ′ whose Lie algebra is h ′ (S). Then, H ′ (S) is a Cartansubgroup of G ′ and one can prove that all the Cartan subgroups are of this form (up to conjugation).For every S ⊆ Ψ ′ st n , we will denote by H ′ S the subgroup of H ′ C given by:H ′ S = c (S) − H ′ (S) c (S) . Assume that r ≤ s . Then, we define Ψ ′ = n e i − e j , ≤ i < j ≤ r + s o , where e i is the linear form on h ′ C = C r + s given by e i ( λ , . . . , λ r + s ) = λ i . In this case, the set Ψ ′ st n is equal to { e t − e r + t , ≤ t ≤ r } . Inparticular, H ′ ( ∅ ) = H ′ and if S t = { e − e r + , . . . , e t − e r + t } , we get:(2) H ′ S t = n h = diag( e iX − X r + , . . . , e iX t − X r + t , e iX t + , . . . , e iX r , e iX + X r + , . . . , e iX t + X r + t , e iX r + t + , . . . , e iX r + s ) , X j ∈ R o . ix a subset S ∈ Ψ ′ st n . We denote by ˇH ′ S the preimage of H ′ S in ˇH ′ C . For every ϕ ∈ C ∞ c ( f G ′ ), we denote by H S ϕ thefunction of ˇH ′ S defined by: H S ϕ (ˇ h ′ ) = ε Ψ ′ S , R (ˇ h ′ )ˇ h ′ P α ∈ Ψ ′ α Y α ∈ Ψ ′ (1 − ˇ h ′− α ) Z G ′ / H ′ (S) ϕ ( g ′ c (S) ˇ p (ˇ h ′ ) c (S) − g ′− ) dg ′ H ′ (S) (ˇ h ′ ∈ ˇH ′ S ) , where Ψ ′ S , R is the subset of Ψ ′ consisting of real roots for H ′ S and ε Ψ ′ S , R is the function defined on ˇH ′ regS by ε Ψ ′ S , R (ˇ h ′ ) = sign Y α ∈ Ψ ′ S , R (1 − ˇ h ′− α ) , To simplify the notations, we denote by ∆ Ψ ′ (ˇ h ′ ) the quantity ∆ Ψ ′ (ˇ h ′ ) = ˇ h ′ P α ∈ Ψ ′ α Y α ∈ Ψ ′ (1 − ˇ h ′− α ) (ˇ h ′ ∈ ˇH ′ S ) . We define ∆ Φ ′ similarly, where Φ ′ = − Ψ ′ . Remark . (1) For every ˇ h ′ ∈ ˇH ′ regS , ∆ Φ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ) = Y α ∈ Ψ ′ + (1 − ˇ h ′ α )(1 − ˇ h ′− α )Note that if S = ∅ , we get for every α ∈ Ψ ′ and ˇ h ′ ∈ ˇH ′ that ˇ h ′ α = ˇ h ′− α . In particular, ∆ Φ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ) = Q α ∈ Ψ ′ (1 − ˇ h ′ α )(1 − ˇ h ′ α ) = Q α ∈ Ψ ′ | − ˇ h ′ α | = | det(Id − Ad(ˇ h ′ )) g ′ / h ′ | . Similarly, if S , ∅ , we get that for every α ∈ Ψ ′ and ˇ h ′ ∈ ˇH ′ , there exists β ∈ Φ ′ , independant on ˇ h ′ , such that ˇ h ′ α = ˇ h ′ β . In particular, we get ∆ Φ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ) = Q α ∈ Ψ ′ | − ˇ h ′ α | .For every ˇ h ′ ∈ ˇH ′ S , we denote by | ∆ G ′ (ˇ h ′ ) | = ∆ Φ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ).(2) One can easily check that two Cartan subalgebras h ′ (S ) and h ′ (S ), with S , S ⊆ Ψ ′ st n , are conjugate if andonly if there exists an element of σ ∈ W sending S ∪ ( − S ) onto S ∪ ( − S ) (see [25, Proposition 2.16]).The Weyl’s integration formula can be written with the previous notations as follows Proposition 3.3 (Weyl’s Integration Formula) . For every ϕ ∈ C ∞ c ( f G ′ ) , we get: (3) Z f G ′ ϕ ( ˜ g ′ ) d ˜ g ′ = X S ∈ Ψ ′ st n m S Z ˇH ′ S ε Ψ ′ S , R (ˇ h ′ ) ∆ Φ ′ (ˇ h ′ ) H S ϕ (ˇ h ′ ) d ˇ h ′ . where m S are complex numbers. Here, the subsets S of Ψ ′ st n are defined up to equivalence (see Remark 3.2).Proof. See [3, Section 2, Page 3830]. (cid:3)
Remark . In particular, if we fix S ⊆ Ψ ′ st n and ϕ ∈ C ∞ c ( f G ′ ) such that supp( ϕ ) ⊆ f G ′ · f H ′ (S) reg , the previousformula can be written as follow: Z f G ′ ϕ ( ˜ g ′ ) d ˜ g ′ = m S Z ˇH ′ S ε Ψ ′ S , R (ˇ h ′ ) ∆ Φ ′ (ˇ h ′ ) H S ϕ (ˇ h ′ ) d ˇ h ′ . et H C , G ′ C ⊆ GL C (W) the complexifications of H and G ′ . We denote by G ′ i C the subgroup of G ′ C consisting ofelements commuting with the element i introduced in Equation (1).As proved in [3, Section 2], the character Θ defined in [1, Definition 4.23] extends to a rational function on e H C · f G ′ i C given by Θ (˜ h ˜ g ′ ) = ( − u det (˜ h ˜ g ′ )det(1 − hg ′ ) (˜ h ∈ e H C , ˜ g ′ ∈ f G ′ i C ) , where 2 u is the maximal dimension of a real subspace of W on which the symmetric form h J · , ·i is negative definite.More precisely, according to [3, Proposition 2.1], we get: Proposition 3.5.
For every ˇ h ∈ ˇH C and ˇ h ′ ∈ ˇH ′ C , we get: det k (ˇ h ) W h ∆ Ψ (ˇ h ) Θ ( ˇ p (ˇ h ) ˇ p (ˇ h ′ )) ∆ Φ (ˇ h ′ ) = X σ ∈ W (H ′ C ) ( − u + α sign( σ ) | W (Z ′ C , H ′ C ) | det − k ( σ − (ˇ h ′ )) W h ∆ Φ (Z ′ ) ( σ − (ˇ h ′ ))det(1 − p ( h ) p ( h ′ )) σ W h , where α ∈ { , − } depends only on the choice of the positive roots Ψ and Φ ′ , k ∈ { , − } is defined byk = − if n ′ − n ∈ Z otherwiseand W h is the set of elements of W commuting with h .Remark . One can easily check that the space W h is given byW h = n X i = Hom(V ′ i , V i ) . For every S ⊆ Ψ ′ st n , we denote by S the subset of { , . . . , r + s } given by S = n j , ∃ α ∈ S such that α (J ′ j ) , o . Let σ ∈ S n ′ and S ⊆ Ψ ′ st n , we denote by Γ σ, S the subset of h ′ defined as(4) Γ σ, S = ( Y ∈ h ′ , h Y · , ·i σ W h ∩ P j < S Hom(V ′ j , V) > ) , and let E σ, S = g exp( i Γ σ, S ) the corresponding subset of e H ′ C , where g exp is a choice of exponential map g exp : h ′ C → e H ′ C obtained by choosing an element e π − { } . Theorem 3.7.
For every ˇ h ∈ ˇH = ˇH ∅ and ϕ ∈ C ( f G ′ ) , we get: det k (ˇ h ) W h ∆ Ψ (ˇ h ) Z f G ′ Θ ( ˇ p (ˇ h ) ˜ g ′ ) ϕ ( ˜ g ′ ) d ˜ g ′ = X σ ∈ W (H ′ C ) X S ⊆ Ψ ′ st n M S ( σ ) lim r ∈ E σ, S r → Z ˇH ′ S det − k ( σ − (ˇ h ′ )) W h ∆ Φ ′ (Z ′ ) ( σ − (ˇ h ′ ))det(1 − p ( h ) rp ( h ′ )) σ W h ε Φ ′ S , R (ˇ h ′ ) H S ( ϕ )(ˇ h ′ ) d ˇ h ′ , where M S ( σ ) = ( − u ε ( σ ) m S | W (Z ′ C , H ′ C ) | . The theorem 3.7 tells us how to compute Chc ˜ h for an element ˜ h in the compact Cartan H = H( ∅ ). Using [2], itfollows that the value of Chc on the other Cartan subgroups can be computed explicitely by knowing how to do itfor the compact Cartan (we will assume, without loss of generality, that p ≤ q , in particular, the number of Cartansubgroups of G, up to conjugation, is p + otation 3.8. For every i ∈ max( p , min( r , s )), we define the set S i S i = { e − e α + , . . . , e i − e α + i } if r ≤ s n e − e β + , . . . , e i − e β + i o otherwise , where α = p if r ≤ pr otherwise and β = p if s ≤ ps otherwise .For every i ∈ [ | , p | ] and j ∈ [ | , min( r , s ) | ], we denote by H(S i ) and H ′ (S j ) the Cartan subgroups of G and G ′ respectively and let H(S i ) = T(S i )A(S i ) and H ′ (S j ) = T ′ (S j )A ′ (S j ) be the decompositions of H(S i ) and H ′ (S j ) asin [26, Section 2.3.6]. In particular, H(S k ) = H ′ (S k ) for every k ∈ [ | , min( p , min( r , s )) | ].To simplify, we assume that r ≤ s . We denote by V , i the subspace of V on which A(S i ) acts trivially and by V , i the orthogonal complement of V , i in V. Let V , i = X i ⊕ Y i be a complete polarization of V , i . We assume thatwe have a natural embedding of V , i into V ′ such that X i ⊕ Y i is a complete polarization with respect to ( · , · ) ′ (i.e i ≤ r ). Let U i be the orthogonal complement of V , i in V ′ ; in particular, we get a natural embedding:GL(X i ) × G(U i ) ⊆ G ′ = U( r , s ) . We denote by T (S i ) the maximal subgroup of T(S i ) which acts trivially on V , i and let T (S i ) the subgroup of T(S i )such that T(S i ) = T (S i ) × T (S i ) with T (S i ) ⊆ G(V , i ). In particular,(5) H(S i ) = T (S i ) × A(S i ) × T (S i ) . Similarly, we get a decomposition of H ′ (S i ) os the form:(6) H ′ (S i ) = T ′ (S i ) × A ′ (S i ) × T ′ (S i ) . Let η (S i ) and η ′ (S i ) be the nilpotent Lie subalgebras of u ( p , q ) and u ( r , s ) respectively given by η (S i ) = Hom(X i , V , i ) ⊕ Hom(X i , Y i ) ∩ u ( p , q ) , η ′ (S i ) = Hom(U i , X i ) ⊕ Hom(X i , Y i ) ∩ u ( r , s ) . We will denote by W , i the subspace of W defined by Hom(U i , V , i ) and by P(S i ) and P ′ (S i ) the parabolic subgroupsof G and G ′ respectively whose Levi factors L(S i ) and L ′ (S i ) are given byL(S i ) = GL(X i ) × G(V , i ) L ′ (S i ) = GL(X i ) × G(U i ) , and by N(S i ) : = exp( η (S i )) and N ′ (S i ) : = exp( η ′ (S i )) the unipotent radicals of P(S i ) and P ′ (S i ) respectively. Remark . One can easily check that the forms on V , i and U i have signature ( p − i , q − i ) and ( r − i , s − i )respectively.As proved in [2, Theorem 0.9], for every ˜ h = ˜ t ˜ a ˜ t ∈ e H(S i ) reg (using the decomposition of H(S i ) given in Equation(5)) and ϕ ∈ C ∞ c ( f G ′ ), we get:(7) | det(Ad(˜ h ) − Id) η (S i ) | Chc ˜ h ( ϕ ) = Cd S i (˜ h ) ε (˜ t ˜ a ) Z GL(X i ) / T (S i ) × A(S i ) Z e G(U i ) ε (˜ t ˜ a ˜ y )Chc W , i (˜ t ˜ y )d ′ S i ( g ˜ t ˜ ag − ˜ y ) ϕ f K ′ f N ′ (S i ) ( g ˜ t ˜ ag − ˜ y ) d ˜ ydg , where C is a constant defined in [2, Theorem 0.9], ε is the character defined in [2, Lemma 6.3], d S i : e L(S i ) → R and d ′ S i : e L ′ (S i ) → R are given byd S i (˜ l ) = | det(Ad(˜ l ) η (S i ) ) | , d ′ S i (˜ l ′ ) = | det(Ad(˜ l ′ ) η ′ (S i ) ) | , (˜ l ∈ e L(S i ) , ˜ l ′ ∈ e L ′ (S i )) , nd ϕ f K ′ f N ′ (S i ) is the Harish-Chandra transform of ϕ , i.e. the function on e L ′ (S i ) defined by: ϕ f K ′ f N ′ (S i ) (˜ l ′ ) = Z f N ′ (S i ) Z f K ′ ϕ (˜ k ˜ l ′ ˜ n ˜ k − ) d ˜ kd ˜ n , (˜ l ′ ∈ e L ′ (S i )) . One can easily check that (G(V , i ) , G(U i )) is an irreducible dual pair in Sp(W , i ) of the same type of G , G ′ . More-over, the element ˜ t is contained in the compact Cartan of G(V , i ). In particular, it follows from Theorem 3.7 thatthe integral Z e G(U i ) ε (˜ t ˜ a ˜ y )Chc W , i (˜ t ˜ y )d ′ S i ( g ˜ t ˜ ag − ˜ y ) ϕ f K ′ f N ′ (S i ) ( g ˜ t ˜ ag − ˜ y ) d ˜ y can be seen as a finite sum of integrals, where the test function ϕ is replaced by ε (˜ y )d ′ S i ( g ˜ t ˜ ag − ˜ y ) ϕ f K ′ f N ′ (S i ) ( g ˜ t ˜ ag − ˜ y ) , ˜ y ∈ e G(U i ). Notation 3.10.
For every j ∈ { , . . . , p } and k ∈ { , . . . , p − j } , we denote by S jk = { e j + − e p + j + , . . . , e j + k − e p + j + k } the subset of Ψ st n ( g ( V , j ) C , t (S j ) C ) and by H(S jk ) the corresponding Cartan subgroup of G(V , j ). By convention,H(S j ) = T (S j ) is the compact Cartan subgroup of G(V , j ).Assume that r ≤ s . For j ∈ { , . . . , r } , and k ∈ { , . . . , r − j } , we denote by S jk = { e j + − e r + j + , . . . , e j + k − e r + j + k } the subset of Ψ ′ st n ( g (U j ) C , t ′ (S j ) C ) and by H ′ (S jk ) the corresponding Cartan subgroup of G(U j ). By convention,H ′ (S j ) = T ′ (S j ) is the compact Cartan subgroup of G(U j ). Remark . Let i ∈ { , . . . , min( p , r ) } and j ≤ i . We denote by X j , Y j , X i , Y i the subspaces of V as before. Thereexists subspaces e X j , i and e Y j , i such that X i = X j ⊕ e X j , i and Y i = Y j ⊕ e X j , i . In particular, U j = U i ⊕ e X j , i ⊕ e Y j , i .The Cartan subgroup H ′ (S i ) is included in Levi factor L ′ (S j ) of the parabolic P ′ (S j ) of G. Let H ′ (S ji − j ) be theCartan subgroup of G(U j ) as in Notation 3.10. As in Equation (5), we have:H ′ (S ji − j ) = T ′ (S ji − j ) × A ′ (S ji − j ) × T ′ (S ji − j ) , and then we get the following decomposition of H ′ (S i ):(8) H ′ (S i ) = T ′ (S i ) × A ′ (S i ) | {z } ⊆ GL(X i ) × T ′ (S i ) |{z} ⊆ G(U i ) = T ′ (S j ) × A ′ (S j ) | {z } ⊆ GL(X j ) × T ′ (S ji − j ) × A ′ (S ji − j ) | {z } ⊆ GL( e X j , i ) × T ′ (S i ) |{z} ⊆ G(U i ) | {z } ⊆ GL( e X j , i ) × G(U i ) ⊆ G(U j ) | {z } ⊆ GL(X j ) × G(U j ) = L ′ (S j ) . Finally, one can see easily that T ′ (S i ) = T ′ (S ji − j ) and then,H ′ (S i ) = T ′ (S j ) × A ′ (S j ) × H ′ (S ji − j ) . We finish this section with a technical lemma which will be useful in Section 6.
Lemma 3.12.
For every f ∈ C ∞ c ( f G ′ ) and ˜ h ′ ∈ f H ′ (S i ) , we get: Z GL(X j ) / T ′ (S j ) × A ′ (S j ) Z G(U j ) / H ′ (S ji − j ) f f K ′ f N ′ (S j ) ( g g ˜ h ′ g − g − ) dg dg = D L ′ (S i ) (˜ h )D L ′ (S j ) (˜ h ) Z GL(X i ) / T ′ (S i ) × A ′ (S i ) Z G(U i ) / T ′ (S i ) f f K ′ f N ′ (S i ) ( g g ˜ h ′ g − g − ) dg dg , here D L ′ (S j ) and D L ′ (S i ) are given by: D L ′ (S j ) (˜ h ′ ) = | det(Id − Ad(˜ h ′ ) − ) l ′ (S j ) / h ′ (S j ) | , D L ′ (S i ) (˜ h ′ ) = | det(Id − Ad(˜ h ′ ) − ) l ′ (S i ) / h ′ (S i ) | . Proof.
As explained in [2, Appendix A], we have: Z f G ′ / f H ′ (S i ) f ( g ˜ hg − ) dg = D L ′ (S i ) (˜ h )D L ′ (S ) (˜ h ) Z L ′ (S i ) / H ′ (S i ) f f K ′ f N ′ (S i ) ( l ′ ˜ hl ′− ) dl ′ = D L ′ (S i ) (˜ h )D L ′ (S ) (˜ h ) Z GL(X i ) / T ′ (S i ) × A ′ (S i ) Z G(U i ) / T (S i ) f f K ′ f N ′ (S i ) ( g g ˜ hg − g − ) dg dg where D L ′ (S ) (˜ h ) = D G ′ (˜ h ′ ) = | det(Ad(˜ h ) − − Id) g ′ / h ′ (S i ) | . Similarly, using that H ′ (S i ) ⊆ P ′ (S j ), we get: Z f G ′ / f H ′ (S j ) f ( g ˜ hg − ) dg = D L ′ (S j ) (˜ h )D L ′ (S ) (˜ h ) Z L ′ (S j ) / H ′ (S i ) f f K ′ f N ′ (S j ) ( l ′ ˜ hl ′− ) dl ′ = D L ′ (S j ) (˜ h )D L ′ (S ) (˜ h ) Z GL(X j ) / T ′ (S j ) × A ′ (S j ) Z G(U j ) / H ′ (S ji − j ) f f K ′ f N ′ (S j ) ( g g ˜ hg − g − ) dg dg , and the lemma follows. (cid:3)
4. T ransfer of invariant eigendistributions
We start this section by recalling the notion of invariant eigendistributions. We keep the notations of AppendixA. Let G be a connected real reductive Lie group, D ′ (G) be the space of distributions of G, i.e. the continuouslinear forms on C ∞ c (G) and D GG (G) the space of bi-invariant di ff erential operators on G as in Notations A.5.For every f ∈ C ∞ c (G) and g ∈ G, we denote by f g the function of C ∞ c (G) defined by f g ( x ) = f ( gxg − ) , x ∈ G. Wesay that T ∈ D ′ (G) is a G-invariant distribution if T ( f g ) = T ( f ) for every f ∈ C ∞ c (G) and g ∈ G. Definition 4.1.
A distribution T on G is an eigendistribution if there exists an algebra homomorphism χ T :D GG (G) → C such that D ( T ) = χ T ( D ) T for every D ∈ D GG (G).As proved by Harish-Chandra (see [9, Theorem 2]), for every invariant eigendistribution T on G, there exists alocally integrable function f T on G which is analytic on G reg such that T = f T , i.e. for every ϕ ∈ C ∞ c (G), T ( ϕ ) = Z G f T ( g ) ϕ ( g ) dg . Remark . (1) Using the isomorphism defined in Appendix A, Theorem A.6, an eigendistribution T is aninvariant distribution such that there exists a character χ T of Z( U ( g C )) such that zT = χ T ( z ) T for every z ∈ Z( U ( g C )).(2) Let ( Π , H ) be a representation of G. Following [6], we say that the representation Π is permissible if Π ( z ) is a scalar multiple of the unit operator for every z ∈ Z(G) ∩ D, where D is the analytic subgroup ofG corresponding to Z( k ) ( k being the Lie algebra of a maximal compact subgroup K of G). A permissi-ble representation is said quasi-simple if there exists an homomorphism χ of Z( U ( g C )) into C such that d Π ( z )( η ) = χ ( z ) η for every z ∈ Z( U ( g C )) and η in the Garding space Gar( Π , H ) (for the definition ofGar( Π , H ), see [6, Part II]). In particular, for such representations, Harish-Chandra proved that for every ϕ ∈ C ∞ c (G), the operator Π ( ϕ ) is a trace class operator (see [7, Section 5]) and the corresponding map Π : C ∞ c (G) ∋ ϕ → tr( Π ( ϕ )) ∈ C is a distribution in the sense of Laurent Schwartz (see [7, Section 5]);the map Θ Π is called the global character of Π . Using that Θ Π is an invariant eigendistribution, it followsthat there exists a locally integrable function Θ Π on G, analytic on G reg , such that Θ Π ( ϕ ) = Z G Θ Π ( g ) ϕ ( g ) dg , ( ϕ ∈ C ∞ c (G)) . The function Θ Π is the character of Π . As proved in [18], every irreducible unitary representation is quasi-simple, in particular, every discrete series representations (see Section 5) has a character, whose value onH reg is explicit (see Theorem 5.4). Notation 4.3.
For every reductive group G, we denote by I (G) the space of orbital integrals on G as in [4,Section 3]. Roughly speaking, the set I (G) is a subset of C ∞ (G reg ) G satisfying 4 conditions (see [4, Pages 579-580]). This space is endowed with a natural topology (see [4, Section 3.3]). We denote by J G the map J G : C ∞ c (G) → I (G) given as follow: for every γ ∈ G reg , there exists a unique, up to conjugation, Cartan subgroupsH( γ ) of G such that γ ∈ H( γ ), and for every ϕ ∈ C ∞ c (G), we define J G ( ϕ )( γ ) by:J G ( ϕ )( γ ) = | det(Id − Ad( γ − )) g / h ( γ ) | Z G / H( γ ) ϕ ( g γ g − ) dg . As proved in [4, Theorem 3.2.1], the map: J G : C ∞ c (G) → I (G)is well-defined and surjective. We denote by I ′ (G) the set of continuous linear forms on I (G) and by J t G : I ′ (G) → D ′ (G) the transpose of J G defined byJ t G ( T )( ϕ ) = T (J G ( ϕ )) , ( T ∈ I ′ (G) , ϕ ∈ C ∞ c (G)) . As proved in [4, Theorem 3.2.1], J t G ( T ) is a G-invariant distribution on G and the corresponding map:J t G : I ′ (G) → D ′ (G) G is bijective.Let (G , G ′ ) be an irreducible dual pair in Sp( W ) such that rk(G) ≤ rk(G ′ ) and ( I ( e G) , J e G ), ( I ( f G ′ ) , J f G ′ ) be thecorresponding space of orbital integrals on e G and f G ′ respectively. To simplify, we assume that both G and G ′ areconnected). For every function ϕ ∈ C ∞ c ( f G ′ ), we denote by Chc( ϕ ) the e G-invariant function on e G reg given by:Chc( ϕ )(˜ h i ) = Chc ˜ h i ( ϕ ) , (˜ h i ∈ e H i reg ) . In [3], the authors proved the following results:
Theorem 4.4.
For every ϕ ∈ C ∞ c ( f G ′ ) , Chc( ϕ ) ∈ I ( e G) and the corresponding map Chc : C ∞ c ( f G ′ ) → I ( e G) is continuous. Moreover, if J f G ′ ( ϕ ) = , we get that Chc( ϕ ) = , i.e. the map Chc : C ∞ c ( f G ′ ) → I ( e G) factorsthrough I ( f G ′ ) and we get a transfer of orbital integrals: Chc : I ( f G ′ ) → I ( e G) . y dualizing the previous map, we get Chc t : I ′ ( e G) → I ′ ( f G ′ ) given byChc t ( τ )( φ ) = τ (Chc( φ )) ( τ ∈ I ′ ( e G) , φ ∈ I ( f G ′ )) . By using the isomorphisms J t e G and J t f G ′ , we get a map Chc ∗ : D ′ ( e G) e G → D ′ ( f G ′ ) f G ′ given by Chc ∗ = J t f G ′ ◦ Chc t ◦ (J t e G ) − .We denote by Eigen( e G) (resp. Eigen( f G ′ )) the set of invariant eigendistributions on e G (resp. f G ′ ). Theorem 4.5.
The map
Chc ∗ : D ′ ( e G) e G → D ′ ( f G ′ ) f G ′ sends Eigen( e G) e G into Eigen( f G ′ ) f G ′ .Remark . If Θ is a distribution on e G given by a locally integrable function Θ on f G ′ , we get for every ϕ ∈ C ∞ c ( f G ′ )the following equality:Chc ∗ ( Θ )( ϕ ) = n X i = | W (H i ) | Z e H i reg Θ (˜ h i ) | det(1 − Ad(˜ h − i )) g / h i | Chc( ϕ )(˜ h i ) d ˜ h i . where H , . . . , H n is a maximal set of non-conjugate Cartan subgroups of G.We recall the following conjecture. Conjecture 4.7.
Let G and G ′ be the Zariski identity components of G and G ′ respectively. Let Π ∈ R ( e G , ω )satisfying Θ Π | e G / f G1 = = O(V), where V is an even dimensional vector space over R or C . Then, up to aconstant, Chc ∗ ( Θ Π ) = Θ Π ′ on f G ′ .In few cases, the conjecture is well-known: if G is compact (see [23]) and if (G , G ′ ) is in the stable range (see[24]). In this paper, we are investigating the case rk(G) = rk(G ′ ), with Π a discrete series representation of e G. Wewill focus our attention on the dual pair of unitary groups satisfying rk(G) = rk(G ′ ), using some results of A. Paulthat we recall in the next section.5. D iscrete series representations and a result of A. P aul
Let G be a connected real reductive Lie group.
Definition 5.1.
We say that an irreducible representation ( Π , ( H , h· , ·i )) is a discrete series representation if all thefunctions τ u , v , u , v ∈ H , are in L (G), where τ u , v : G ∋ g → h g ( u ) , v i ∈ C . Remark . One can prove that the condition given in the previous definition is equivalent to say that the repre-sentation ( Π , H ) is equivalent with a direct summand of the right regular representation of G on L (G).Moreover, as recalled in [15, Section 9.3], for such a representation ( Π , H ), there exists a positive number d Π (depending on the Haar measure dg on G), called the formal degree of Π , such that for every u , u , v , v ∈ H , Z G h Π ( g ) u , v ih Π ( g ) u , v i dg = h u , u ih v , v i d Π . In his papers [8] and [10], Harish-Chandra gave a classification of the discrete series representations of G. First ofall, he proved that G has discrete series if and only if G has a compact Cartan subgroup (see [10, Theorem 13]).Let K be a maximal compact subgroup of G and H a Cartan subgroup of K. He also proved that the set of discreteseries is indexed by a lattice of i h ∗ . We say few words about this now. Let Ψ = Ψ ( g C , h C ) be the set of roots of g , Ψ ( k ) = Ψ ( k C , h C ) be the set of compact roots of g , ρ = P α ∈ Ψ + α and ρ ( k ) = P α ∈ Φ + ( k ) α . otation 5.3. For every g ∈ G, we denote by D g the function on R given byD g ( t ) = det(( t + g − Ad( g )) ( t ∈ R ) . In particular, D g ( t ) = n P i = t i D i ( g ), with n = dim(G). The D ′ i s are analytic on G and let l be the least integer such thatD l ,
0. The integer l is the rank of g . We denote by D( g ) the coe ffi cient of t l in the previous polynomial and byG reg the set of g ∈ G such that D( g ) , Theorem 5.4.
Let λ be an element of i h ∗ such that λ + ρ is analytic integral. Then, there exists a discrete seriesrepresentation ( Π λ , H λ ) of G such that:(1) The representation Π λ has infinitesimal character χ λ as in Remark A.9,(2) The linear form ν = λ + ρ − ρ ( k ) is the highest weight of the lowest K -type for Π λ | K and the multiplicity ofthe corresponding representation Π ν in Π λ | K is one.The parameter λ is called the Harish-Chandra parameter of Π λ . Moreover, if we denote by Θ λ the distributioncharacter of Π and by Θ λ the corresponding locally integrable function on G reg , we get that the restriction of Θ λ of Π to H reg is given by the following formula Θ λ (exp( X )) = ( − dim(G) − dim(K)2 X w ∈ W ( k ) ε ( w ) e ( w λ )( X ) Q α> ( e α ( X )2 − e − α ( X )2 ) , ( X ∈ h reg ) . Remark . As proved in [8], for every discrete series Π of G with Harish-Chandra parameter λ , we get:sup g ∈ G reg | D( g ) | | Θ λ ( g ) | < ∞ . The previous properties of Θ λ characterize the discrete series characters inside the space of invariant distributionsof G. More precisely, as proved in [8, Lemma 44], we have the following result. Theorem 5.6.
Let Θ λ be G -invariant distribution on G such that:(1) z Θ λ = γ ( λ )( z ) Θ λ , z ∈ Z( U ( g C )) ,(2) sup g ∈ G reg | D( g ) | | Θ λ ( g ) | < ∞ ,(3) Θ λ = pointwise on H reg .Then, Θ λ = . The previous theorem will be central for us in Section 6 to prove the conjecture 4.7 for discrete series representa-tions in the equal rank case. We now recall a key result of A. Paul for unitary groups. Let (G , G ′ ) = (U( p , q ) , U( r , s ))be a dual pair of unitary groups in Sp(2( p + q )( r + s ) , R ). As explained in [19, Section 1.2], the double cover of e U( p , q ) is isomorphic to(9) e U( p , q ) ≈ n ( g , ξ ) ∈ U( p , q ) × C ∗ , ξ = det( g ) r − s o . In particular, all the genuine admissible representations of e U( p , q ) are the form Π ⊗ det r − s , where det r − s is thegenuine character of e U( p , q ) given by det r − s ( g , ξ ) = ξ and Π is an admissible representation of U( p , q ). From nowon, we fix p and q and let r and s vary under the condition that p + q = r + s . In particular, under this condition, itfollows from Equation (9) that the double cover U( p , q ) stays the same when r and s vary.In [19, Section 6], A. Paul proved the following theorem: heorem 5.7. For every genuine irreducible admissible representation ( Π , H Π ) of e U( p , q ) , there exists a uniquepair of integers ( r , s ) = ( r Π , s Π ) such that p + q = r + s with θ r , s ( Π ) , . She also obtained more precise results for discrete series representations (see [19, Theorem 6.1] or [20, Theo-rem 2.7]).
Notation 5.8.
We fix a basis { e , . . . , e n } of h ∗ . In particular, every linear form λ on h can be written as λ = n P i = λ i e i or also as λ = ( λ , . . . , λ n ). Theorem 5.9.
Let Π be a discrete series representation of e U( p , q ) , the corresponding representation θ r Π , s Π ( Π ) is adiscrete series representation of e U( r Π , s Π ) .More precisely, if the Harish-Chandra parameter of Π is of the form λ = λ a , b = ( α , . . . , α a , β , . . . , β p − a , γ , . . . , γ b , δ , . . . , δ q − b ) , with α i , β j , γ k , δ l ∈ Z + such that α > . . . > α a > > β > . . . > β p − a and γ > . . . > γ b > > δ > . . . > δ q − b ,then ( r Π , s Π ) = ( a + q − b , b + p − a ) and the corresponding Harish-Chandra parameter λ ′ = λ ′ a , b of θ r Π , s Π ( Π ) is ofthe form: λ ′ a , b = ( α , . . . , α a , δ , . . . , δ q − b , γ , . . . , γ b , β , . . . , β p − a ) .
6. P roof of C onjecture for discrete series representations in the equal rank case In this section, we are interested in the dual pair (G , G ′ ) = (U( p , q ) , U( r , s )) such that p + q = r + s . Withoutloss of generality, we assume that p ≤ q . In particular, the number of Cartan subgroups of G, up to conjugation, is p +
1. We denote by n = p + q . Let ( V = C p + q , ( · , · )) and ( V ′ = C r + s , ( · , · ) ′ ) be the hermitian and skew-hermitianspaces corresponding to G and G ′ respectively. In this case, the space W = Hom( V ′ , V ) = M(( r + s ) × ( p + q ) , C )and for every w ∈ W , there exists a unique element w ∗ ∈ Hom( V , V ′ ) = M(( p + q ) × ( r + s ) , C ) such that: (cid:0) w ( v ′ ) , v (cid:1) = (cid:0) v ′ , w ∗ ( v ) (cid:1) ′ , ( v ∈ V , v ′ ∈ V ′ ) . One can prove that w ∗ = i Id p , q w t Id r , s and the symplectic form h· , ·i on W h w , w ′ i = Re(tr( w ′∗ w )) = − Im(tr(Id p , q w ′ t Id r , s w )) ( w , w ′ ∈ W ) . Let V i = V ′ i = C e i . The subspaces h and h ′ of g and g ′ respectively given by: h = h ′ = { y = ( iX , . . . , iX n ) , X i ∈ R } are Cartan subalgebras. Moreover, we get: W h = n M i = Hom( V i , V ′ i ) = n M i = i R E i , i . Let Π be a discrete series of e U( p , q ), Θ Π be the corresponding element of D ′ ( e G) e G , Θ Π the corresponding locallyintegrable function on e G such that Θ Π = T Θ Π and χ Π the infinitesimal character of Π .As recalled in Theorem 4.5, Chc ∗ ( Θ Π ) is an element of Eigen( f G ′ ) f G ′ . According to [9, Theorem 2], the distribution Θ ′ Π = Chc ∗ ( Θ Π ) is given by a locally integrable function Θ ′ Π on f G ′ , analytic on f G ′ reg . Notation 6.1.
From now on, we fix an element f − π − ( {− } ). Let c : g c → G c the Cayleytransform, where g c and G c are defined as in Section 2. As explained in [21, Lemma 3.5], there exists a uniquesmooth map ˜ c : g c → e G c such that π ◦ ˜ c = c and ˜ c (0) = f − heorem 6.2. The value of Θ ′ Π on the compact Cartan f H ′ = f H ′ ( ∅ ) is given by the following formula: ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = C | W (H) | X σ ∈ S r + s ε ( σ )det ( σ (ˇ h ′ )) W h ′ lim r → r ∈ E σ, ∅ Z ˇH Θ Π ( ˇ p (ˇ h )) ∆ (ˇ h )det (ˇ h )det(1 − p (ˇ h ) rp (ˇ h ′ )) σ W h d ˇ h (ˇ h ′ ∈ ˇH ′ reg ) , where H = H( ∅ ) is the compact Cartan of G and C = χ Π ( f − Θ ( f − − u .Proof. Let ϕ be a function in C ∞ c ( f G ′ ). According to Remark 4.6, we get that Θ ′ Π ( ϕ ) = p X i = | W (H(S i )) | Z e H(S i ) reg Θ Π (˜ h i ) | det(1 − Ad(˜ h − i )) g / h i | Chc( ϕ )(˜ h i ) d ˜ h i . where H(S i ) is a set of Cartan subgroups as in Remark 2, and let H = H( ∅ ) the compact Cartan of G. Now, if weassume that supp( ϕ ) ⊆ f G ′ · f H ′ , then Θ ′ Π ( ϕ ) = | W (H) | Z e H reg Θ Π (˜ h ) | det(1 − Ad(˜ h − )) g / h | Chc( ϕ )(˜ h ) d ˜ h According to [3, Equation 8] and Theorem 3.7, we get: Θ ′ Π ( ϕ ) = | W (H) | Z e H reg Θ Π (˜ h ) | det(1 − Ad(˜ h − i )) g / h | Chc( ϕ )(˜ h ) d ˜ h = ( − u C | W (H) | Z ˇH reg Θ Π ( ˇ p (ˇ h )) | ∆ G (ˇ h ) | Z f G ′ Θ ( ˇ p (ˇ h ) ˜ g ′ ) ϕ ( ˜ g ′ ) d ˜ g ′ ! d ˇ h = ( − u + C | W (H) | Z ˇH reg Θ Π ( ˇ p (ˇ h )) ∆ Ψ (ˇ h )det (ˇ h ) det − (ˇ h ) ∆ Ψ (ˇ h ) Z f G ′ Θ ( ˇ p (ˇ h ) ˜ g ′ ) ϕ ( ˜ g ′ ) d ˜ g ′ ! d ˇ h = − C m | W (H) | X σ ∈ S r + s ε ( σ ) lim r → r ∈ E σ, ∅ Z ˇH reg Θ Π ( ˇ p (ˇ h )) ∆ Ψ (ˇ h )det (ˇ h ) Z ˇH ′ det ( σ − (ˇ h ′ )) W h det(1 − p (ˇ h ) rp (ˇ h ′ )) σ W h H ∅ ( ϕ )(ˇ h ′ ) d ˇ h ′ d ˇ h With such assumptions on the support of ϕ , we get using Equation (3) Θ ′ Π ( ϕ ) = Z f G ′ Θ ′ Π ( ˜ g ′ ) ϕ ( ˜ g ′ ) d ˜ g ′ = m Z ˇH ′ ∆ Ψ ′ (ˇ h ′ ) H ∅ ( Θ ′ Π ϕ )(ˇ h ′ ) d ˇ h ′ = − m Z ˇH ′ Θ ′ Π ( ˇ p (ˇ h ′ )) ∆ Ψ ′ (ˇ h ′ ) H ∅ ( ϕ )(ˇ h ′ ) d ˇ h ′ By identifications, we get, up to a constant, that: ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = C | W (H) | X σ ∈ S r + s ε ( σ )det ( σ − (ˇ h ′ )) W h lim r → r ∈ E σ, ∅ Z ˇH reg Θ Π ( ˇ p (ˇ h )) ∆ Ψ (ˇ h )det (ˇ h )det(1 − p (ˇ h ) rp ( σ (ˇ h ′ ))) W h d ˇ h and the theorem follows. (cid:3) We know that the set of roots for ( g , h ) is given by n ± ( e i − e j ) , ≤ i < j ≤ n o . Let K = U( p ) × U( q ) be a maximal compact subgroup of G. Let Ψ ( k ) = Ψ ( k C , h C ) be a set of compact positive rootsgiven by: Ψ ( k ) = n e i − e j , ≤ i < j ≤ p o ∪ n e i − e j , p + ≤ i < j ≤ n o . he compact Weyl group W ( k ) = W (K , H) is S p × S q . Let λ = p + q P i = λ i e i be the Harish-Chandra parameter of Π .Using Theorem 5.4, the value of Θ Π on e H reg is given by: Θ Π ( ˇ p (ˇ h )) = ( − α p , q X β ∈ S p × S q ε ( β ) ( β ˇ h ) λ Q α> (ˇ h α − ˇ h − α ) , (ˇ h ′ ∈ ˇH reg ) , with α p , q = dim(G) − dim(K)2 = pq . Using that W h = n L i = Hom( V ′ i , V i ), we get:det(1 − p ( h ) rp ( h ′ )) σ W h = n Y i = (cid:16) − h i ( rh ′ ) − σ ( i ) (cid:17) = ( − n n Y i = ( rh ′ ) − σ ( i ) n Y i = (cid:0) h i − ( rh ′ ) σ ( i ) (cid:1) , and det ( σ − (ˇ h ′ )) W h = n Y i = h ′− σ ( i ) , det (ˇ h ) W h = n Y i = h i . To simplify the notations, we will denote by ξ the element of h ∗ C given by ξ = n P i = e i . We recall a basic Cauchyintegral formula. Lemma 6.3.
Let k ∈ Z and a ∈ C ∗ \ S . Then, i π Z S z k z − adz = a k if k ≥ and | a | < − a k if k < and | a | > otherwise For every ˇ h ′ ∈ ˇH ′ reg , we get from Theorem 6.2: ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = e C X σ ∈ S r + s X β ∈ S p × S q ε ( σ ) ε ( β ) n Y i = h ′− σ ( i ) n Y i = h ′ σ ( i ) lim r → r ∈ E σ, ∅ Z ˇH ( β ˇ h ) λ n Q i = h in Q i = ( h i − ( rh ′ ) σ ( i ) ) d ˇ h = e C X σ ∈ S r + s X β ∈ S p × S q ε ( σ ) ε ( β ) n Y i = h ′ σ ( i ) lim r → r ∈ E σ, ∅ Z ˇH ( β ˇ h ) λ + ξ n Q i = ( h i − ( rh ′ ) σ ( i ) ) d ˇ h = e C X σ ∈ S r + s X β ∈ S p × S q ε ( σ ) ε ( β ) n Y i = h ′ σ ( i ) lim r → r ∈ E σ, ∅ Z H n Q i = h λ i + β − ( i ) n Q i = ( h i − ( rh ′ ) σ ( i ) ) dh = e C n Q i = h ′ i (2 i π ) n X σ ∈ S r + s X β ∈ S p × S q ε ( σ ) ε ( β ) lim r → r ∈ E σ, ∅ n Y i = Z S z λ i − z − ( rh ′ ) σ ( β − ( i )) dz , where e C = ( − pq C W (H) . emma 6.4. For every σ ∈ S r + s , the space E σ, ∅ is given by E σ, ∅ = h ′ = ( e − X , . . . , e − X n ) ∈ H ′ C , X σ ( i ) > if i ∈ { , . . . , p } and σ ( i ) ∈ { , . . . , r } X σ ( i ) < if i ∈ { , . . . , p } and σ ( i ) ∈ { r + , . . . , r + s } X σ ( i ) < if i ∈ { p + , . . . , p + q } and σ ( i ) ∈ { , . . . , r } X σ ( i ) > if i ∈ { p + , . . . , p + q } and σ ( i ) ∈ { r + , . . . , r + s } Proof.
Let w = n P i = w i E i , i ∈ W h , σ ∈ S r + s and y = ( iX , . . . , iX n ) ∈ h ′ , with X j ∈ R . Then, h y ( σ ( w )) , σ ( w ) i = h y n X i = w i E i ,σ ( i ) , n X j = w j E j ,σ ( j ) i = −h n X i = w i y σ ( i ) E i ,σ ( i ) , n X j = w j E j ,σ ( j ) j i = n X i = n X j = Im(tr( w j Id p , q E σ ( j ) , j Id r , s w i y σ ( i ) E i ,σ ( i ) )) = n X i = Im(tr(Id p , q w i E i ,σ ( i ) Id r , s w i y σ ( i ) E σ ( i ) , i )) = p X i = Im(tr( | w i | y σ ( i ) E i ,σ ( i ) Id r , s E σ ( i ) , i )) − n X i = p + Im(tr( | w i | y σ ( i ) E i ,σ ( i ) Id r , s E σ ( i ) , i )) = p X i = σ ( i ) ∈{ ,..., r } | w i | X σ ( i ) − p X i = σ ( i ) ∈{ r + ,..., n } | w i | X σ ( i ) − n X i = p + σ ( i ) ∈{ ,..., r } | w i | X σ ( i ) + n X i = p + σ ( i ) ∈{ r + ,..., n } | w i | X σ ( i ) In particular, using Equation (4), we get: Γ σ, ∅ = y = ( iX , . . . , iX n ) ∈ h ′ , X σ ( i ) > i ∈ { , . . . , p } and σ ( i ) ∈ { , . . . , r } X σ ( i ) < i ∈ { , . . . , p } and σ ( i ) ∈ { r + , . . . , n } X σ ( i ) < i ∈ { p + , . . . , n } and σ ( i ) ∈ { , . . . , r } X σ ( i ) > i ∈ { p + , . . . , n } and σ ( i ) ∈ { r + , . . . , n } The result follows using that E σ, ∅ = exp( i Γ σ, ∅ ). (cid:3) Proposition 6.5.
Let Π ∈ R ( e U( p , q ) , ω ) be a discrete series representation of Harish-Chandra parameter λ a , b as inTheorem 5.9 and let ( r , s ) = ( r Π , s Π ) the unique integers such that θ r , s ( Π ) , . The value of Θ ′ Π on f H ′ reg = ] H ′ ( ∅ ) reg is given by ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = − n − a − b ε ( τ a , b ) e C X σ ∈ S r × S s ε ( σ )( σ ˇ h ′ ) τ a , b λ a , b , where τ a , b ∈ S r + s is defined by: • If r ≤ p, τ a , b = ( a + , p + b + a + , p + b + . . . ( r , p + q ) , • If p + ≤ r ≤ p + b, τ a , b ∈ Stab S r + s ( { , . . . , a } ∪ { r + , . . . , p + b } ) and satisfies: τ a , b ( a + = p + b + , . . . , τ a , b ( r ) = r + s , τ a , b ( p + b + = a + , . . . , τ a , b ( p + q ) = r . If r ≥ p + b + , τ a , b ∈ Stab S r + s ( { , . . . , a } ∪ { p + b + , . . . , r } ) and satisfies τ a , b ( a + = r + , . . . , τ a , b ( p + b ) = r + s , τ a , b ( r + = a + , . . . , τ a , b ( r + s ) = p + b + . Notation 6.6.
For every subset { i , . . . , i k } of { , . . . , p } (resp. { p + , . . . , p + q } , { , . . . , r } or { r + , . . . , r + s } ), wedenote by { i , . . . , i k } c the set { , . . . , p } \ { i , . . . , i k } (resp. { p + , . . . , p + q } \ { i , . . . , i k } , { , . . . , r } \ { i , . . . , i k } or { r + , . . . , r + s } \ { i , . . . , i k } ).For two subsets { a , . . . , a w } and { b , . . . , b w } of { , . . . , p + q } , we denote by S { b ,..., b w }{ a ,..., a w } the groups of bijectionsbetween { a , . . . , a w } and { b , . . . , b w } .Similarly, for every β ∈ S p × S q , we denote by S ( β ) { b ,..., b w }{ a ,..., a w } the groups of bijections between { β ( a ) , . . . , β ( a w ) } and { b , . . . , b w } . Obviously, S p × S q = [ { i ,..., i a }⊆{ ,..., p } [ { j ,..., j b }⊆{ p + ,..., p + q } S { i ,..., i a } c { ,..., p − a } × S { i ,..., i a }{ p − a + ,..., p } × S { j ,..., j b } c { p + ,..., p + q − b } × S { j ,..., j b }{ p + q − b + ,..., p + q } for every 1 ≤ t ≤ p . Proof.
To simplify the notations, we will denote by R( σ, λ a , b , β ) , σ ∈ S p + q , β ∈ S p × S q , the following term:R( σ, λ a , b , β ) = lim r → r ∈ E σ, ∅ n Y i = Z S z λ i − z − ( rh ′ ) σ ( β − ( i )) dz According to Lemmas 6.3 and 6.4, we get that R( σ, λ a , b , β ) , σ ◦ β − ∈ [ { i ,..., i q − b }⊆{ ,..., r } [ { j ,..., j p − a }⊆{ r + ,..., r + s } S { i ,..., i q − b } c { ,..., a } × S { j ,..., j p − a }{ a + ,..., p } × S { j ,..., j p − a } c { p + ,..., p + b } × S { i ,..., i q − b }{ p + b + ,..., p + q } . We first assume that r ≤ p . In this case, using that { a + , . . . , p } = { a + , . . . , r } ∪ { r + , . . . , p } , we get S r × S s = [ { i ,..., i q − b }⊆{ ,..., r } [ { j ,..., j p − a }⊆{ r + ,..., r + s } S { i ,..., i q − b } c { ,..., a } × S { j ,..., j p − a }{ a + ,..., p } × S { j ,..., j p − a } c { p + ,..., p + b } × S { i ,..., i q − b }{ p + b + ,..., p + q } ◦ σ , where σ = ( a + , p + b + a + , p + b + . . . ( r , p + q ). For every β ∈ S p × S q , there exists exactly r ! s ! elementsin σ ∈ S r + s such that σ ◦ β − ∈ S r × S s ◦ σ .Then, X σ ∈ S r + s X β ∈ S p × S q ε ( σβ ) lim r → r ∈ E σ, ∅ p + q Y i = Z S z λ i − z − ( rh ′ ) σ ( β − ( i )) dz = ( − p + q − a − b (2 i π ) p + q p ! q ! X τ ∈ S r × S s ◦ σ ε ( τ ) p + q Y i = h ′ λ i − τ ( i ) = ( − p + q − a − b (2 i π ) p + q p ! q ! X τ ∈ S r × S s ε ( τσ − ) p + q Y i = h ′ λ i − τ ( σ − ( i )) = ( − p + q − a − b (2 i π ) p + q p ! q ! X τ ∈ S r × S s ε ( τσ − )( σ τ − ˇ h ′ ) λ a , b − ξ = ( − p + q − a − b (2 i π ) p + q p ! q ! ε ( σ ) X τ ∈ S r × S s ε ( τ )( τ ˇ h ′ ) σ ( λ a , b − ξ ) , where ξ = n P i = e i , i.e. ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = − p + q − a − b ε ( σ ) e C X τ ∈ S r × S s ε ( τ )( τ ˇ h ′ ) σ ( λ a , b ) . Now assume that r > p . We distinguish two cases. If p + ≤ r ≤ p + b , then, S r × S s = [ { i ,..., i q − b }⊆{ ,..., r } [ { j ,..., j p − a }⊆{ r + ,..., r + s } S { i ,..., i q − b } c { ,..., a } × S { j ,..., j p − a }{ a + ,..., p } × S { j ,..., j p − a } c { p + ,..., p + b } × S { i ,..., i q − b }{ p + b + ,..., p + q } ◦ η, or every η ∈ S r + s satisfying η { , . . . , a } = { , . . . , a } , η { r + , . . . , p + b } = { r + , . . . , p + b } , η { a + , . . . , r } ⊆ { p + b + , . . . , r + s } and η { p + b + , . . . , p + q } ⊆ { a + , . . . , r } . Let σ be the element of Stab S r + s ( { , . . . , a } ∪ { r + , . . . , p + b } )given by σ ( a + = p + b + , . . . , σ ( r ) = r + s , σ ( p + b + = a + , . . . , σ ( p + q ) = r . This element satisfy the previous conditions and we get: ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = − p + q − a − b ε ( σ ) e C X τ ∈ S r × S s ε ( τ )( τ ˇ h ′ ) σ ( λ a , b ) . Similarly, r ≥ p + b + S r × S s = [ { i ,..., i q − b }⊆{ ,..., r } [ { j ,..., j p − a }⊆{ r + ,..., r + s } S { i ,..., i q − b } c { ,..., a } × S { j ,..., j p − a }{ a + ,..., p } × S { j ,..., j p − a } c { p + ,..., p + b } × S { i ,..., i q − b }{ p + b + ,..., p + q } ◦ η, for every η ∈ S r + s satisfying η { , . . . , a } = { , . . . , a } , η { p + b + , . . . , r } = { p + b + , . . . , r } , η { a + , . . . , p + b } ⊆ { r + , . . . , r + s } and η { r + , . . . , r + s } ⊆ { a + , . . . , p + b + } . Let σ be the element of Stab S r + s ( { , . . . , a } ∪ { p + b + , . . . , r } )given by σ ( a + = r + , . . . , σ ( p + b ) = r + s , σ ( r + = a + , . . . , σ ( r + s ) = p + b + . This element satisfies the previous conditions and we get: ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( ˇ p (ˇ h ′ )) = − p + q − a − b ε ( σ ) e C X τ ∈ S r × S s ε ( τ )( τ ˇ h ′ ) σ ( λ a , b ) . (cid:3) Proposition 6.7.
For every Π ∈ R ( e U( p , q ) , ω ) , we get sup ˜ g ′ ∈ f G ′ reg | D( ˜ g ′ ) | | Θ ′ Π ( ˜ g ′ ) | < ∞ . We first need to introduce some notations.
Notation 6.8.
Let k ∈ [ | , min( r , s ) | ], we denote by η ′ S k = Ad( c (S k ) − )( η ′ (S k )) ⊆ η ′ + C , where η ′ + C = L α ∈ Ψ ′ + g ′ C α , where g ′ C α is the eigenspace corresponding to α ∈ Ψ ′ .By keeping the notations of Section 3, we get that Ψ ′ can be decomposed as follow:(10) Ψ ′ = Ψ ′ ( gl (X k )) ∪ Ψ ′ ( g (U k )) ∪ Ψ ′ ( η ′ (S k )) , ( k ∈ [ | , min( p , min( r , s )) | ]) . Finally, we denote by W ( g (U k )) the Weyl group corresponding to ( g (U k ) C , t (S k ) C ). Lemma 6.9.
For every ˜ h ′ ∈ f H ′ (S k ) reg , det(Id − Ad(˜ h ′ )) η ′ (S k ) ∈ R ∗ + .Proof. To make things easier, we will consider S k = { e − e r + s − k + , . . . , e k − e r + s } . As in Equation (2), we have:H ′ S k = c (S k ) − H ′ (S k ) c (S k ) = n h ′ = ( e i θ − X , . . . , e i θ k − X k , t , . . . , t r + s − k , e i θ + X , . . . , e i θ k + X k ) , t i ∈ U(1) , θ i , X i ∈ R o . We denote by h ′ = c (S k ) − h ′ c (S k ). Obviously, we get:det(Id − Ad(˜ h ′ )) η ′ (S k ) = det(Id − Ad( h ′ )) η ′ (S k ) = det(Id − Ad( h ′ )) η ′ S k = det(Id − Ad(ˇ h ′ )) η ′ S k . We know that det(Id − Ad( h )) η ′ S k = Y α ∈ Ψ ′ ( η ′ (S k )) (1 − h − α ) , nd that Ψ ′ ( η ′ (S k )) = { e i − e j , ≤ i ≤ k , k + ≤ j ≤ r + s − k } ∪ { e i − e j , k + ≤ i ≤ r + s − k , r + s − k + ≤ j ≤ r + s } . Let α = e i − e j , with 1 ≤ i ≤ k , k + ≤ j ≤ r + s − k and let α = e j − e r + s − i + . Then,(1 − h α )(1 − h α ) = (1 − e i θ i − X i t − j − k )(1 − t j − k e − i θ i − X i ) = | − e i θ i − X i t − j − k | , and the result follows. (cid:3) Proof of Proposition 6.7.
Without loss of generality, we can assume that r ≤ s . We distinguish two cases. Wefirst start with p ≤ r . Note that in this case, H S i = H ′ S i (resp. H(S i ) = H ′ (S i )) for every 0 ≤ i ≤ p , withS i = { e − e r + , . . . , e i − e r + i } as in Notation 3.8.In this case, we get, using [2, Corollary A.4], that for every ϕ ∈ C ∞ c ( f G ′ ): Θ ′ Π ( ϕ ) = Z f G ′ Θ ′ Π (˜ g ′ ) ϕ (˜ g ′ ) d ˜ g ′ = r X i = m i Z ˇH ′ S i ε S i , R (ˇ h ′ ) ∆ Φ ′ (ˇ h ′ ) H S i ( Θ ′ Π ϕ )(ˇ h ′ ) d ˇ h ′ = r X i = m i Z ˇH ′ S i ε S i , R (ˇ h ′ ) ∆ Φ ′ (ˇ h ′ ) ε S i , R (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ) Z G ′ / H ′ (S i ) Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) ϕ ( g ′ c (S i ) ˇ p (ˇ h ′ ) c (S i ) − g ′− ) dg ′ ! d ˇ h ′ = r X i = m i Z ˇH S i Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) | ∆ G ′ (ˇ h ′ ) | Z G ′ / H ′ (S i ) ϕ ( g ′ c (S i ) ˇ p (ˇ h ′ ) c (S i ) − g ′− ) dg ′ d ˇ h ′ = m Z ˇH ′ S0 Θ ′ Π ( ˇ p (ˇ h ′ )) | ∆ G ′ (ˇ h ′ ) | Z G ′ / H ′ (S ) ϕ ( g ′ ˇ p (ˇ h ′ ) g ′− ) dg ′ d ˇ h ′ + p X i = m i Z ˇH ′ S i Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) | ∆ G ′ (ˇ h ′ ) | Λ ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) Z L ′ (S i ) / H ′ (S i ) ϕ f K ′ f N ′ (S i ) ( l ′ c (S i ) ˇ p (ˇ h ′ ) c (S i ) − l ′− ) dl ′ d ˇ h ′ (11) + r X i = p + m i Z ˇH ′ S i Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) | ∆ G ′ (ˇ h ′ ) | Z L ′ (S p ) / H ′ (S i ) ϕ f K ′ f N ′ (S p ) ( g ′ c (S i ) ˇ p (ˇ h ′ ) c (S i ) − g ′− ) dg ′ d ˇ h ′ where | ∆ G ′ (ˇ h ′ ) | = ∆ Φ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ) as in Remark 3.2, L ′ (S i ) is defined in Section 3 and Λ ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) is givenby: Λ ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) = D L ′ (S i ) ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − )D L ′ (S ) ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) = | det(Id − Ad( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) − ) l ′ (S i ) / h ′ (S i ) | | det(Id − Ad( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) − ) g ′ / h ′ (S i ) | . Using that Θ ′ Π = Chc ∗ ( Θ Π ), we get: Θ ′ Π ( ϕ ) = p X j = Z e H(S j ) Θ Π (˜ h ) | det(Id − Ad(˜ h ) − ) g / h (S j ) | Chc ˜ h ( ϕ ) d ˜ h . Using Equation (7), we get that: Θ ′ Π ( ϕ ) = p X j = Z f T (S j ) Z e A(S j ) Z f T (S j ) Θ Π (˜ t ˜ a ˜ t ) | det(Id − Ad(˜ t ˜ a ˜ t ) − ) g / h (S j ) | Chc ˜ t ˜ a ˜ t ( ϕ ) d ˜ t d ˜ ad ˜ t = p X j = C j Z f T (S j ) Z e A(S j ) Z f T (S j ) Θ Π (˜ t ˜ a ˜ t ) | det(Id − Ad(˜ t ˜ a ˜ t ) − ) g / h (S j ) | ε (˜ t ˜ a ˜ y )d S j (˜ t ˜ a ˜ t ) | det(Id − Ad(˜ t ˜ a ˜ t ) − ) η (S j ) | (12) Z GL(X j ) / T ′ (S j ) × A ′ (S j ) Z e G(U j ) Chc W , j (˜ t ˜ y ) ε (˜ t ˜ a ˜ y )d ′ S j ( g ˜ t ˜ ag − ˜ y ) ϕ f K ′ N ′ (S j ) ( g ˜ t ˜ ag − ˜ y ) d ˜ ydgd ˜ t d ˜ ad ˜ t et i ∈ [ | , r | ] and ϕ ∈ C ∞ c ( f G ′ ) such that supp( ϕ ) ⊆ f G ′ · f H ′ (S i ). On one hand, using Equation (11), we get: Θ ′ Π ( ϕ ) = m i Z ˇH ′ S i Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) | ∆ G ′ (ˇ h ′ ) | Λ ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) Z L ′ (S i ) / H ′ (S i ) ϕ f K ′ f N ′ (S i ) ( l ′ c (S i ) ˇ p (ˇ h ′ ) c (S i ) − l ′− ) dl ′ d ˇ h ′ = m i Z ˇT ′ , S i Z ˇA ′ S i Z ˇT ′ , S i Θ ′ Π ( c (S i ) ˇ p (ˇ t ′ ˇ a ′ ˇ t ′ ) c (S i ) − ) | ∆ G ′ (ˇ t ′ ˇ a ′ ˇ t ′ ) | Λ ( c (S i ) ˇ p (ˇ t ′ ˇ a ′ ˇ t ′ ) c (S i ) − ) Z GL(X i ) / T ′ (S i ) × A ′ (S i ) Z G(U i ) / T ′ (S i ) ϕ f K ′ f N ′ (S i ) ( g g c (S i ) ˇ p (ˇ t ′ ˇ a ′ ˇ t ′ ) c (S i ) − g − g − ) dg dg d ˇ t ′ d ˇ a ′ d ˇ t ′ . In particular, for every j < i , we get from Equation (8): Θ ′ Π ( ϕ ) = m i Z ˇT ′ , S j Z ˇA ′ S j Z ˇT ′ , S ji − j Z ˇA ′ S ji − j Z ˇT ′ , S i Θ ′ Π ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − ) | ∆ G ′ (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) | Λ ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − ) Z GL(X i ) / T ′ (S i ) × A ′ (S i ) Z G(U i ) / T ′ (S i ) ϕ f K ′ f N ′ (S i ) ( g g c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − g − g − ) dg dg d ˇ t j d ˇ a j d ˇ h j d ˇ b j d ˇ t i . On the other hand, it follows from Equation (12) that Θ ′ Π ( ϕ ) = min( i , p ) X j = C i Z f T (S j ) Z e A(S j ) Z f T (S j ) Θ Π (˜ t ˜ a ˜ t ) | det(Id − Ad(˜ t ˜ a ˜ t ) − ) g / h (S j ) | ε (˜ t ˜ a )d S j (˜ t ˜ a ˜ t ) | det(Id − Ad(˜ t ˜ a ˜ t )) η (S j ) | Z GL(X j ) / T ′ (S j ) × A ′ (S j ) Z e G(U j ) Chc W , j (˜ t ˜ y ) ε (˜ t ˜ a ˜ y )d ′ S j ( g ˜ t ˜ ag − ˜ y ) ϕ f K ′ f N ′ (S j ) ( g ˜ t ˜ ag − ˜ y ) d ˜ ydgd ˜ t d ˜ ad ˜ t = i X j = C j Z ˇT , S j Z ˇA S j Z ˇT , S j Θ Π ( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) | det(Id − Ad( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) − ) g / h (S j ) | ε ( c (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − )d S j ( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) | det(Id − Ad( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − )) η (S j ) | Z GL(X j ) / T ′ (S j ) × A ′ (S j ) Z e G(U j ) Chc W , j ( ˇ p (ˇ t )˜ y ) ε ( c (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − ˜ y )d ′ S j ( gc (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − g − ˜ y ) ϕ f K ′ e N ′ (S j ) ( gc (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − g − ˜ y ) d ˜ ydgd ˇ t d ˇ ad ˇ t = min( i , p ) X j = C j Z ˇT , S j Z ˇA S j Z ˇT , S j Θ Π ( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) | det(Id − Ad( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) − ) g / h (S j ) | ε ( c (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − )d S j ( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) | det(Id − Ad( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − )) η (S j ) | X σ ∈ W ( g (U j )) ε ( σ ) lim r → r ∈ E σ, S ji − j Z GL(X j ) / T ′ (S j ) × A ′ (S j ) det(ˇ t ) W t , S j ∆ Ψ ′ ( g (U j )) (ˇ t ) − Z ˇH ′ S ji − j det − ( σ − (ˇ h ′ )) W t , S j det(1 − p (ˇ h ′ ) rp (ˇ t )) σ W t , S j Z e G(U j ) / H ′ (S ji − j ) ∆ Ψ ′ + ( g (U j )) (ˇ h ′ ) ε ( c (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − g c (S ji − j ) ˇ p (ˇ h ′ ) c (S ji − j ) − g − )d ′ S j ( g c (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − g − g c (S ji − j ) ˇ p (ˇ h ′ ) c (S ji − j ) − g − ) ϕ f K ′ e N ′ (S j ) ( g c (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − g − g c (S ji − j ) ˇ p (ˇ h ′ ) c (S ji − j ) − g − ) dg d ˇ h ′ dg d ˇ t d ˇ ad ˇ t Using Lemma 3.12 and the equality c (S j ) c (S ji − j ) = c (S i ), we get: Θ ′ Π ( ϕ ) = min( i , p ) X j = C j Z ˇT , S j Z ˇA S j Z ˇT , S j Θ Π ( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) | det(Id − Ad( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) − ) g / h (S j ) | ε ( c (S j ) ˇ p (ˇ t ˇ a ) c (S j ) − )d S j ( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − ) | det(Id − Ad( c (S j ) ˇ p (ˇ t ˇ a ˇ t ) c (S j ) − )) η (S j ) | X σ ∈ W ( g (U j )) ε ( σ ) lim r → r ∈ E σ, S ji − j Z GL(X i ) / T ′ (S i ) × A ′ (S i ) det(ˇ t ) W t , S j ∆ Ψ ′ ( g (U j )) (ˇ t ) − Z ˇH ′ S ji − j det − ( σ − (ˇ h ′ )) W t , S j ∆ Ψ ′ + ( g (U j )) (ˇ h ′ )det(1 − p (ˇ h ′ ) rp (ˇ t )) σ W t , S j D L ′ (S i ) ( c (S i ) ˇ p (ˇ t ˇ a ˇ h ′ ) c (S i ) − )D L ′ (S j ) ( c (S i ) ˇ p (ˇ t ˇ a ˇ h ′ ) c (S i ) − ) Z e G(U i ) / T ′ (S i ) ε ( c (S i ) ˇ p (ˇ t ˇ a ˇ h ′ ) c (S i ) − )d ′ S j ( g g c (S i ) ˇ p (ˇ t ˇ a ˇ h ′ ) c (S i ) g − g − ) ϕ f K ′ f N ′ (S j ) ( g g c (S i ) ˇ p (ˇ t ˇ a ˇ h ′ ) c (S i ) − g − g − ) dg d ˇ h ′ dg d ˇ t d ˇ ad ˇ t . s explained in Remark 3.11, for every 0 ≤ j ≤ i , we have the following decomposition H ′ (S i ) = T ′ (S j ) × A ′ (S j ) × T ′ (S ji − j ) × A ′ (S ji − j ) × T ′ (S i ). In particular, every element h ∈ H ′ (S i ) can be written as h = t j a j h j b j t i . We get similarresults for H ′ S i . In particular, we get that the value of Θ ′ Π on H ′ (S i ) is given by: Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) | ∆ G ′ (ˇ h ′ ) | Λ ( c (S i ) ˇ p (ˇ h ) c (S i ) − ) = min( i , p ) X j = C j X σ ∈ W ( g (U j )) ε ( σ ) ∆ Ψ ′ + ( g (U j )) (ˇ h j ˇ b j ˇ t i )det − ( σ − (ˇ h j ˇ b j ˇ t i )) W t , S j D L ′ (S i ) ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − )D L ′ (S j ) ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − ) ε ( c (S j ˇ p (ˇ h j ˇ b j ˇ t i ) c (S j ) − )lim r → r ∈ E σ, S ji − j Z ˇT , S j d S j ( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − ) Θ Π ( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − ) | det(Id − Ad( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − ) − ) g / h (S j ) | det(ˇ h ) W t , S j ∆ Ψ ′ ( g (U j )) (ˇ h )det(Id − Ad( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − )) η (S j ) det(1 − p (ˇ h j ˇ b j ˇ t i ) rp (ˇ h )) σ W t , S j d ˇ h . Using Equation (10), we getD L ′ (S i ) ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − ) = Y α ∈ Ψ ′ + ( l ′ (S i )) | − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α | = Y α ∈ Ψ ′ + ( gl ′ (X i )) | − (ˇ t j ˇ a j ˇ h j ˇ b j ) α | Y α ∈ Ψ ′ + ( g (U i )) | − (ˇ t i ) α | andD L ′ (S j ) ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − ) = Y α ∈ Ψ ′ + ( l ′ (S j )) | − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α | = Y α ∈ Ψ ′ + ( gl ′ (X j )) | − (ˇ t j ˇ a j ) α | Y α ∈ Ψ ′ + ( g (U j )) | − (ˇ h j ˇ b j ˇ t i ) α | , so in particularD L ′ (S ) ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − )D L ′ (S j ) ( c (S i ) ˇ p (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) c (S i ) − ) | ∆ G ′ (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) | = Q α ∈ Ψ ′ + | − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α | Q α ∈ Ψ ′ + ( gl ′ (X j )) | − (ˇ t j ˇ a j ) α | Q α ∈ Ψ ′ + ( g (U j )) | − (ˇ h j ˇ b j ˇ t i ) α | . Finally, ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) = min( i , p ) X j = C j X σ ∈ W ( g (U j )) ε ( σ ) ∆ Ψ ′ + ( g (U j )) (ˇ h j ˇ b j ˇ t i )det − ( σ − (ˇ h j ˇ b j ˇ t i )) W t , S j ∆ Ψ ′ (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) ε ( c (S j ˇ p (ˇ h j ˇ b j ˇ t i ) c (S j ) − ) Q α ∈ Ψ ′ + | − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α | Q α ∈ Ψ ′ + ( gl (X j )) | − (ˇ t j ˇ a j ) α | Q α ∈ Ψ ′ + ( g (U j )) | − (ˇ h j ˇ b j ˇ t i ) α | lim r → r ∈ E σ, S ji − j Z ˇT , S j (cid:16) Θ Π ( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − ) ∆ Ψ ′ (ˇ t j ˇ a j ˇ h ) (cid:17) ∆ Φ ′ (ˇ t j ˇ a j ˇ h )det(ˇ h ′ ) W t , S j ∆ Ψ ′ ( g (U j )) (ˇ h )det(Id − Ad( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − )) η (S j ) det(1 − p (ˇ h j ˇ b j ˇ t i ) rp (ˇ h )) σ W t , S j d ˇ h Again, we get from Equation (10) that ∆ Ψ ′ (ˇ t j ˇ a j ˇ h ) = ∆ Ψ ′ ( gl (X i ) (ˇ t j ˇ a j ) ∆ Ψ ′ ( g (U j )) (ˇ h ) ∆ Ψ ′ ( η ′ (S j )) (ˇ t j ˇ a j ˇ h ) = (ˇ t j ˇ a j ) − ρ ′ ( gl (X i )) − ρ ′ ( η ′ (S j )) ˇ h − ρ ′ ( η ′ (S j )) Y α ∈ Ψ ′ + ( gl (X j )) (1 − (ˇ t j ˇ a j ) α ) Y α ∈ Ψ ′ + ( η ′ (S j )) (1 − (ˇ t j ˇ a j ˇ h ) α ) ∆ Ψ ′ ( g (U j )) (ˇ h ) , then, ∆ Ψ ′ (ˇ h ′ ) Θ ′ Π ( c (S i ) ˇ p (ˇ h ′ ) c (S i ) − ) = min( i , p ) X j = X σ ∈ W ( g (U j )) ε ( σ )det − ( σ − (ˇ h j ˇ b j ˇ t i )) W t , S j ε ( c (S j ˇ p (ˇ h j ˇ b j ˇ t i ) c (S j ) − ) α ∈ Ψ ′ + − ˇ h ′ α Q α ∈ Ψ ′ + | − ˇ h ′ α | Q α ∈ Ψ ′ + ( gl (X i )) − (ˇ t j ˇ a j ) α Q α ∈ Ψ ′ + ( gl (X i )) | − (ˇ t j ˇ a j ) α | Q α ∈ Ψ ′ + ( g (U j )) − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α Q α ∈ Ψ ′ + ( g (U j )) | − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α | lim r → r ∈ E σ, S ji − j Z ˇT , S j (cid:16) Θ Π ( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − ) ∆ Ψ ′ (ˇ t j ˇ a j ˇ h ) (cid:17) det(ˇ h ) W t , S j det(1 − p (ˇ h j ˇ b j ˇ t i ) rp (ˇ h )) σ W t , S j d ˇ h Using [15, Theorem 10 35] and [15, Theorem 10.48], it follows that Θ Π ( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − ) ∆ Ψ ′ (ˇ t j ˇ a j ˇ h ) = X w ∈ S p + q c ( w )(ˇ t j ˇ a j ˇ h ) w λ = X w ∈ S p + q c ( w )(ˇ t j ˇ a j ) w λ ˇ h w λ , where c ( w ) are complex numbers. Using that ε ( c (S j ˇ p (ˇ h j ˇ b j ˇ t i ) c (S j ) − ) Q α ∈ Ψ ′ + − ˇ h ′ α Q α ∈ Ψ ′ + | − ˇ h ′ α | Q α ∈ Ψ ′ + ( gl (X i )) − (ˇ t j ˇ a j ) α Q α ∈ Ψ ′ + ( gl (X i )) | − (ˇ t j ˇ a j ) α | Q α ∈ Ψ ′ + ( g (U j )) − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α Q α ∈ Ψ ′ + ( g (U j )) | − (ˇ t j ˇ a j ˇ h j ˇ b j ˇ t i ) α | is of norm 1, it follows from Lemma 6.3 and Theorem 5.6 thatlim r → r ∈ E σ, S ji − j Z ˇT , S j (cid:16) Θ Π ( c (S j ) ˇ p (ˇ t j ˇ a j ˇ h ) c (S j ) − ) ∆ Ψ ′ (ˇ t j ˇ a j ˇ h ) (cid:17) det(ˇ h ) W t , S j det(1 − p (ˇ h j ˇ b j ˇ t i ) rp (ˇ h )) σ W t , S j d ˇ h is a finite sum of bounded exponentials, and then, for every i ,sup ˜ h ′ ∈ e H ′ (S i ) | D(˜ h ′ ) | | Θ ′ Π (˜ h ′ ) | < ∞ . The proof is similar if r ≤ p . Note that in this case, H S i = H ′ S i (resp. H(S i ) = H ′ (S i )) for every 0 ≤ i ≤ r , withS i = { e − e p + , . . . , e i − e p + i } , and in particular, the Cauchy-Harish-Chandra integrals Chc ˜ h i , ˜ h i ∈ H(S i ) , r + ≤ i ≤ p ,does not give any contribution to Θ ′ Π on the di ff erent Cartan subgroups e H ′ (S k ), 0 ≤ k ≤ r . It concludes the proof ofthe proposition. (cid:3) Lemma 6.10.
Let (G = U( p , q ) , G ′ = U( r , s )) , p + q = r + s and Π ∈ R ( e U( p , q ) , ω ) a discrete series representationwith Harish-Chandra parameter λ a , b as in Theorem 5.9. For every z ∈ Z( U ( gl ( r + s , C ))) , we get:z Θ ′ Π = χ λ ′ a , b ( z ) Θ ′ Π , where χ λ ′ a , b = λ ′ a , b ( γ ( z )) as in Appendix A, Remark A.9.Proof. Obviously, in the equal rank case, U ( g C ) = U ( g ′ C ). It follows from [3, Theorem 1.4] that Chc ∗ ( z Θ Π ) = z Chc( Θ Π ). Because λ a , b and λ ′ a , b are conjugated under S r + s , the result follows from Theorem A.10. (cid:3) Corollary 6.11.
For every discrete series representation Π of e U( p , q ) , we get Chc ∗ ( Θ Π ) = C Θ θ r , s ( Π ) . with C ∈ C .Proof. Using Theorem 5.6, it follows from Propositions 6.5 and 6.7 and Lemma 6.10 that, up to a scalar, Θ ′ Π = Chc ∗ ( Θ Π ) is either the character of a discrete series representations of e U( r , s ) with Harish-Chandra parameter τ a , b ( λ a , b ) if ( r , s ) = ( r Π , s Π ) or 0 if ( r , s ) , ( r Π , s Π ). The result follows from Theorem 5.4 because τ a , b ( λ a , b ) and λ ′ a , b (as in Theorem 5.9) are conjugated under S r × S s . (cid:3) orollary 6.12. If (G , G ′ ) = (U( p , q ) , U( r , s )) , p + q = r + s and Π ∈ R ( e G , ω ) a discrete series representation of e G . Then, the conjecture 4.7 holds.Proof. It follows from Theorem 7.2 because Π ′ = Π ′ . (cid:3)
7. A commutative diagram and a remark on the distribution Θ Π ′ We start this section by recalling a result of T. Przebinda (see [22]). Let (G , G ′ ) = (G(V , ( · , · ) , G(V ′ , ( · , · ) ′ ))) bean irreducible reductive dual pair in Sp(W). As proved in [14] (see also Section 2), the representations appearing inthe correspondence are realized as quotients of H ∞ , the set of smooths vectors of the metaplectic representation( ω, H ). Let Π ∈ R ( e G , ω ), Π ′ the corresponding element of R ( f G ′ , ω ) and N( Π ⊗ Π ′ ) the ω ∞ ( e G · f G ′ )-invariantsubspace of H ∞ such that Π ⊗ Π ′ ≈ H ∞ / N( Π ⊗ Π ′ ) as in Section 2.In particular,( Π ⊗ Π ′ ) ∗ ≈ ( H ∞ / N( Π ⊗ Π ′ )) ∗ ≈ Ann(N( Π ⊗ Π ′ )) = (cid:8) α ∈ H ∞∗ , α ( X ) = , ( ∀ X ∈ N( Π ⊗ Π ′ )) (cid:9) ⊆ H ∞∗ , i.e. there exists a unique element, up to a constant, Γ Π ⊗ Π ′ ∈ Hom( H ∞ , H ∞∗ ) such that Hom e G · f G ′ ( ω, Π ⊗ Π ′ ) = C · Γ Π ⊗ Π ′ . Remark . Let W = X ⊕ Y be a complete polarization of W. It is well-known that we can realize the representation ω on H = L (X): this is the Schrodinger model. Moreover, the space of smooth vectors of ω is the Schwartzspace S(X) of X.Using the isomorphisms K : S ∗ (W) → S ∗ (X × X) and Op : S ∗ (X × X) → Hom(S(X) , S ∗ (X)) (see [1, Equa-tions (143) and (146)]), there exists a unique distribution f Π ⊗ Π ′ ∈ S ∗ (W) such that Γ Π ⊗ Π ′ = Op ◦ K ( f Π ⊗ Π ′ ). Thedistribution f Π ⊗ Π ′ is called the intertwining distribution corresponding to Π ⊗ Π ′ .As explained in [17, Section 2], the situation turns out to be slightly easier when dim(V) ≤ dim(V ′ ) and ( Π , H Π )a discrete series representation of e G. Under those hypothesis, the space H ∞ ⊗ H ∞ Π has a natural structure of e G-modules. Using the scalar products on H and H Π , we get a natural inner product h· , ·i on H ∞ ⊗ H ∞ Π . Wedenote by h· , ·i Π the following form on H ∞ ⊗ H ∞ Π : h Φ , Φ ′ i Π = Z e G h Φ , ( ω ⊗ Π )( g ) Φ ′ i dg , ( Φ , Φ ′ ∈ H ∞ ⊗ H ∞ Π ) . One can easily prove that in this context, the previous integral converges absolutely. We denote by R( Π ) the radicalof the form h· , ·i Π and we still denote by h· , ·i Π the non-degenerate form we got on H( Π ) = H ∞ ⊗ H ∞ Π / R( Π ). Thegroup f G ′ acts naturally on H( Π ) and we denote by θ ( Π ) the corresponding f G ′ -module. Theorem 7.2 ([17], Section 2) . (1) There exists u , v ∈ H ∞ and x , y ∈ H ∞ Π such that Z e G h u ⊗ x , ( ω ⊗ Π )( ˜ g )( v ⊗ y ) i Π , . Moreover, we get: Z e G |h ω ( ˜ g ) u , v i| dg < + ∞ , ( u , v ∈ H ∞ ) . (2) The representation Π can be embedded in ω as an irreducible subrepresentation and θ ( Π ) defines anirreducible unitary representation on the completion of H( Π ) (completion with respect to h· , ·i Π ).(3) The map Π → θ ( Π ∗ ) coincide with Howe’s duality correspondence. e get the following proposition. Proposition 7.3.
Let (G , G ′ ) = (U(V) , U(V ′ )) with dim(V) ≤ dim(V ′ ) and Π be a discrete series representation of e G . The intertwining distribution is given by f Π ⊗ Π ′ = T( Θ Π ) , where T( Θ Π ) = Z e G Θ Π ( ˜ g )T( ˜ g ) d ˜ g . Proof.
As explained in [22, Theorem 3.1], the previous Lemma follows if the following condition Z e G | Ω ( ˜ g ) || Θ Π ( ˜ g ) | d ˜ g < ∞ is satisfied, where Ω is defined in Appendix B. Using Lemma B.1, it follows that there exists C Ω > Z e G | Ω ( ˜ g ) || Θ Π ( ˜ g ) | d ˜ g ≤ C Ω Z e G Ξ ( ˜ g ) | Θ Π ( ˜ g ) | d ˜ g . Using the fact that every discrete series satisfies the strong inequality (see [26, Section 5.1.1]), it follows from [26,Lemma 5.1.3] that Z e G Ξ ( ˜ g ) | Θ Π ( ˜ g ) | d ˜ g < ∞ , and the proposition follows. (cid:3) Corollary 7.4.
Assume that (G , G ′ ) = (U( p , q ) , U( r , s )) , with p + q = r + s, and let Π be a discrete series repre-sentation of e G . Then, there exists a constant C Π ⊗ Π ′ ∈ C such that T( Θ Π ) = C Π ⊗ Π ′ T(Chc ∗ ( Θ Π )) .Proof. The proof of this corollary follows from Corollary 6.11 and Proposition 7.3. (cid:3)
We finish this section with a remark concerning the global character Θ Π ′ , Π ′ = θ ( Π ) and Π ∈ R ( e G , ω ) a discreteseries representation. We proved in Corollary 6.11 that Chc ∗ ( Θ Π ) = Θ Π ′ if rk(G) = rk(G ′ ). But the global character Θ Π ′ can be obtained via Θ Π in a di ff erent way.As before, we assume that rk(G) ≤ rk(G ′ ). In particular, every discrete series representation Π ∈ R ( e G , ω ) is asub-representation of ω and let H ( Π ) be the Π -isotypic component of H . As explained in Proposition 7.3, T( Θ Π )is well-defined. Moreover, using [1, Section 4.8], the operator ω ( Θ Π ) is a well-defined operator of H and one cancheck that P Π : = ω ( d Π Θ Π ) is a projection operator onto H ( Π ). As a e G × f G ′ -modules, we get H ( Π ) = Π ⊗ Π ′ .Let λ be the Harish-Chandra parameter of Π and ν the lowest e K-type of Π | e K . In particular, according to Theorem5.4, as a e K × f G ′ , we get: H ( Π ) = M ξ ∈ e K Π m ξ Π ξ ⊗ Π ′ = Π ν ⊗ Π ′ ⊕ M ξ , ν ξ ∈ e K Π m ξ Π ξ ⊗ Π ′ , where Π ξ is a e K-module of highest weight ξ and e K Π is the set of irreducible representations of e K such thatHom e K ( H ξ , H ) , { } . We denote by H ( Π )( ν ) the ν -isotypic component of H ( Π ). We denote by P ν : H ( Π ) → H ( Π )( ν ) the corresponding projection operator and let P Π ,ν = P ν ◦ P Π . Clearly, P Π ,ν = Π ( d ν Θ Π ν ) ◦ ω ( d Π Θ Π ) = ω ( d ν Θ Π ν ) ◦ ω ( d Π Θ Π ). In particular, for every ϕ ∈ C ∞ c ( f G ′ ), we get: Θ Π ′ ( ϕ ) = d ν tr(Id H ν ⊗ Π ′ ( ϕ )) = d ν tr( P Π ,ν ◦ ω ( ϕ )) , .e. Θ Π ′ ( ϕ ) = d Π tr Z e K Z e G Z f G ′ Θ Π ν (˜ k ) Θ Π ( ˜ g ) ϕ ( ˜ g ′ ) ω (˜ k ˜ g ˜ g ′ ) d ˜ g ′ d ˜ gd ˜ k ! . In particular, if rk(G) = rk(G ′ ), we get using Corollary 6.11 that:(13) p X i = | W (H i ) | Z e H i Θ Π (˜ h i ) | det(Id − Ad(˜ h i ) − ) g / h i | Chc ˜ h i ( ϕ ) d ˜ h i = d Π tr Z e K Z e G Z f G ′ Θ Π ν (˜ k ) Θ Π ( ˜ g ) ϕ ( ˜ g ′ ) ω (˜ k ˜ g ˜ g ′ ) d ˜ g ′ d ˜ gd ˜ k ! . A ppendix A. S ome standard isomorphisms
A.0.1.
Universal envelopping algebra of g as di ff erential operators on G . Let M be a real connected manifold ofdimension n . We denote by C ∞ (M) the space of smooth functions and C ∞ c (M) the space of compactly supportedfunction on C ∞ (M).We denote by X (M) the set of derivations of C ∞ (M), i.e. X (M) = { X : C ∞ (M) → C ∞ (M) , X ( f g ) = X ( f ) g + f X ( g ) } . The space X (M) is the set of C ∞ -vectors fields of M. Definition A.1.
A continuous endomorphism D of C ∞ c (M) is called a di ff erential. operator if whenever U is anopen set in M and f a function of C ∞ c (M). vanishing on U, then D f vanishes on U.
Proposition A.2.
Let D be a di ff erential operator on M .For each p ∈ M and each open connected neighbourhood U of p on which the local coordinates system Ψ : x → ( x , . . . , x n ) is valid, there exists a finite set of functions a α of class C ∞ such that for each f ∈ C ∞ c (M) with support contained in U , D f ( x ) = P α = ( α ,...,α n ) a α ( x )D α f ◦ Ψ − ( x ) if x ∈ U0 otherwiseProof. The proof of this result can be found in [11, Proposition 1]. (cid:3)
Notation A.3.
We denote by D(M) the set of di ff erential operators.From now on, we assume that G = M is a connected Lie group. We denote by (L , L (G , dg )) the left regularrepresentation. Obviously, the space C ∞ c (G) is G-invariant.We define an action of G on D(G) by( τ ( g )D)( f ) = L g ◦ D( f ◦ L g − ) ( g ∈ G , f ∈ C ∞ c (G) , D ∈ D(G)) . Definition A.4.
We say that D ∈ D(G) is left-invariant if τ ( g )D = D for all g ∈ G, i.e. L g ◦ D( f ) = D( f ◦ L g ).We denote by D G (G) the set of left-invariant di ff erential operators of G. Similarly, we say that D is right invariantif τ ( g )(D) = D for every g ∈ G, where( τ ( g )D)( f ) = R g − ◦ D( f ◦ R g ) ( f ∈ C ∞ c (G) . The operator D is is said to be bi-invariant if τ ( g ) τ ( h )(D) = D for every g , h ∈ G, i.e. R h − ◦ L g ◦ D(L g − ◦ f ◦ R h ) = D( f ) for every f ∈ C ∞ c (G). otation A.5. We denote by D G (G) the set of right-invariant di ff erential operators and by D GG (G) the set of bi-invariant di ff erential operators.We recall the following result. Theorem A.6.
The natural embedding g → D G (G) extends to an algebra isomorphism U ( g ) → D G (G) . Moreover, its restriction to Z( U ( g )) is isomorphic D GG (G) .Proof. The proof of this result can be found in [12]. (cid:3)
A.0.2.
Harish-Chandra isomorphism.
Let g be a complex reductive Lie algebra and h be a Cartan subalgebra of g . We denote by W = W ( g , h ) the corresponding Weyl group.We denote by η + the subalgebra of g given by η + = P α ∈ Ψ + ( g , h ) C X α , where g α = C X α . Similarly, we denote by N and P the following subspaces of U ( g ) given by: N = X α ∈ Φ + ( g , h ) Y α U ( g ) P = X α ∈ Φ + ( g , h ) X α U ( g ) . where Y α is a basis of g − α . Lemma A.7.
We get the following decomposition: (14) U ( g ) = U ( h ) ⊕ ( P + N ) . We denote by p : U ( g ) → U ( h ) the natural projection corresponding to Equation (14). We restrict this map toZ( U ( g )). We denote by ζ the map: ζ : h ∋ h → ζ ( h ) = h − ρ ( h ) . ∈ S( h ) , where ρ = P α ∈ Φ + ( g , h ) α ∈ h ∗ .Using the universal property, we can extend the map ζ to S( h ). We denote by γ the map: γ = ζ ◦ p : Z( U ( g )) → S( h ) . Theorem A.8.
The map γ is an algebra homomorphism which is injective. Moreover, Im( γ ) = S( h ) W and then: γ : Z( U ( g )) → S( h ) W . is a bijection.Remark A.9 . Harish-Chandra’s isomorphism classify all the possible infinitesimal character. Indeed, let λ : h → C be a linear map. Using he universal property of the symmetric algebra [5, Appendix C], the linear form λ can beextended to a linear map λ : S( h ) → C and by using the map λ , we get a map χ λ : Z( U ( g )) → C given by: χ λ ( z ) = λ ( γ ( z )) ( z ∈ Z( U ( g ))) . e recall the following theorem. Theorem A.10.
Let g be a complex reductive Lie algebra and h a Cartan subalgebra of g . Then every homomor-phism of Z( U ( g )) into C sending to is of the form χ λ , λ ∈ h ∗ . If λ and λ ′ ∈ h ∗ , then χ λ = χ λ ′ if and only if λ and λ ′ are in the same W -orbit.In particular, Spec(Z( U ( g ))) ≈ h ∗ / W . The proof of this result can be found in [16].A ppendix
B. A general lemma for unitary groups
Let U be a maximal compact subgroup of Sp(W). It is well-known that the restriction of ω to e U is a direct sumof irreducible representations whose multiplicity is one. Moreover, the lowest e U-type V ω is one-dimensional. Let x be a non-zero vector in V ω and let Ω be the function on f Sp(W) given by Ω ( ˜ g ) = h ω ( ˜ g ) x , x i , ( ˜ g ∈ f Sp(W)) . We denote by ξ ω the (unitary) character of e K such that ω (˜ k ) x = ξ ω (˜ k ) x , ˜ k ∈ e K. One can check that for every˜ k , ˜ k ∈ e K and ˜ g ∈ e G, Ω (˜ k ˜ g ˜ k ) = h ω (˜ k ˜ g ˜ k ) x , x i = h ω ( ˜ g ˜ k ) x , ω (˜ k − ) x i = ξ ω ( k k − ) h ω ( ˜ g ) x , x i = ξ ω ( k k − ) Ω ( ˜ g ) . In particular, the map e G ∋ ˜ g → | Ω ( ˜ g ) | ∈ C is e K-bi-invariant. In particular, using the decomposition f Sp( W ) = e K e A e Kas in [26, Section 3.6.7], with A = Cl(A + ), A + = exp( a + ), a the maximal split Cartan subalgebra of sp (W) and a + = { H ∈ a , α ( H ) > , α ∈ Ψ + } , we get for every ˜ g = ˜ k ˜ a ˜ k ∈ e K e A e K that | Ω ( ˜ g ) | = | Ω (˜ a ) | .We denote by Ξ the e K-bi-invariant function defined in [26, Section 4.5.3].
Lemma B.1.
Let (G , G ′ ) = (U(V) , U(V ′ )) ⊆ Sp((V ⊗ V ′ ) R ) be a dual pairs of unitary groups such that dim C (V) ≤ dim C (V ′ ) . Then, there exists a constant C Ω > such that: (15) | Ω ( ˜ g ) | ≤ C Ω Ξ ( ˜ g ) , ( ˜ g ∈ e G) . Remark
B.2 . As explained in [22, Equation 6.4], for an irreducible reductive dual pair (G , G ′ ), there exist a constantC = C d , d ′ > | Ω (˜ c (X)) | = C | det R (Id − X) | d ′ | det( i Id − JX) | − d ′ , (X ∈ g c ) , where d = dim D (V) and d ′ = dim D (V ′ ). Proof.
We start by determining the value of Ω for the dual pair (G , G ′ ) = (U(1 , , U(1)). Let G = KAK be thedecomposition of G as in [26, Section 3.6.7]. In this case,A = ch( X ) sh( X )sh( X ) ch( X ) , X ∈ R ∗ + . Let a ( X ) ∈ A c and b ( X ) ∈ a c such that a ( X ) = c ( b ( X )). One can easily check that b ( X ) = α ( X ) α ( X ) 0 , where α ( X ) = sh( X )ch( X ) −
1. Note that α ( X ) = X ). et B = { e , e } be a basis of V such that Mat B ( · , · ) = Id , . Then, using that B R = { e , e , ie , ie } is a basis ofthe real vector space V R , it follows that:det R (Id − b ( X )) = det − α ( X ) 0 0 − α ( X ) 1 0 00 0 1 − α ( X )0 0 − α ( X ) 1 = (1 − α ( X ) ) . Similarly, using that J = Mat B R ( · , · ) R = − − , we get:det( i Id − J b ( X )) = det − −
11 0 0 00 1 0 0 − − − α ( X ) 0 0 α ( X ) 0 0 00 0 0 α ( X )0 0 α ( X ) 0 = det − −
11 0 0 00 1 0 0 − α ( X )0 0 − α ( X ) 00 − α ( X ) 0 0 α ( X ) 0 0 0 = det − − α ( X )0 0 α ( X ) − α ( X ) 0 0 − α ( X ) 1 0 0 = (1 + α ( X ) ) Using that th( X ) − X ) − = − X ) , it follows from Remark B.2 that there exists C > | Ω (˜ c ( b ( X ))) | = C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − α ( X ) + α ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Cch( X ) . In particular, for the dual pair (G , G ′ ) = (U(1 , , U( n )), we get for every a ( X ) = c ( b ( X )) ∈ A c ⊆ G that: | Ω (˜ c ( b ( X ))) | = Cch( X ) n . From [26, Theorem 4.5.3], we know that ˜ a ( X ) − ρ ≤ Ξ (˜ a ( X )), with ˜ a ( X ) = ˜ c ( b ( X )). In our case, we get that˜ a ( X ) − ρ = e − X and using that for every n ≥
1, ch( X ) n ≥ ch( X ) ≥ e X , it follows that: | Ω (˜ c ( b ( X ))) | = Cch( X ) n ≤ Cch( X ) ≤ C e − X ≤ C Ξ (˜ a ( X )) . In particular, using the e K-bi-invariance of Ω and Ξ , we get that Ω ( ˜ g ) ≤ C Ξ ( ˜ g ) for every ˜ g ∈ e G for (G , G ′ ) = (U(1 , , U( n )). One can easily check that G ′ can be replaced by U( r , s ) and the computations are similar. e can now extend it to (G , G ′ ) = (U( p , p ) , U( n )). In this case,A = D = D ( X ) D ( X )D ( X ) D ( X ) , X ∈ R ∗ + p where for X = ( X , . . . , X p ), D ( X ) = diag(ch( X ) , . . . , ch( X p )) and D ( X ) = diag(sh( X ) , . . . , sh( X p )). One caneasily check that there exists C > | Ω (˜ c ( b ( X ))) | = C p Q i = ch( X i ) n . In this case, ρ = p P i = p − i + e i , and from Equation (2), we get˜ a ( X ) − ρ = diag( e − X , . . . , e − X p , e X , . . . , e X p ) − ρ = p Y k = e − pX k If n ≥ p , it follows that ch( X k ) n ≥ ch( X k ) p ≥ e pX k for every k ∈ [ | , p | ] and then, | Ω (˜ c ( b ( X ))) | = C p Q i = ch( X i ) n ≤ C p Y k = e − pX k ≤ Ξ (˜ a ( X )) . Again, U( n ) can be replaced by U( r , s ) as long as r + s ≥ p . Finally, it G = U( p , q ), with p ≤ q , we get that:A = D = D ( X ) D ( X )D t ( X ) D ⋄ ( X ) , X ∈ R ∗ + p where D ( X ) = diag(ch( X ) , . . . , ch( X p )), D ( X ) = (diag(sh( X ) , . . . , sh( X p )) , p , q ), where 0 p , q is the zero matrix ofMat( p , q − p ), and D ⋄ ( X ) = diag(ch( X ) , . . . , ch( X p )) 0 p , q − p q − p , p q − p , q − p . and one can check that the computations done for U( p , p ) extends easily to U( p , q ). The lemma follows. (cid:3) Remark
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