Characters of irreducible unitary representations of U(n, n+1) via double lifting from U(1)
aa r X i v : . [ m a t h . R T ] F e b CHARACTERS OF IRREDUCIBLE UNITARY REPRESENTATIONS OF U( n , n + VIA DOUBLELIFTING FROM
U(1)
ALLAN MERINOA bstract . In this paper, we obtained character formulas of irreducible unitary representations of U( n , n +
1) by usingHowe’s correspondence and the Cauchy–Harish-Chandra integral. The representations of U( n , n +
1) we are dealing withare obtained from a double lifting of a representation of U(1) via the dual pairs (U(1) , U(1 , , , U( n , n +
1. I ntroduction
For a finite dimensional representation ( Π , V) of a group G, one can associate a function Θ Π on G given by Θ Π : G ∋ g → tr( Π ( g )) ∈ C . The function Θ Π is the character of the representation ( Π , V), and it determines entirely the representation. Obvi-ously, if we remove the assumption that V is finite dimensional, the map Θ Π does not necessarily makes sense ingeneral. In [6, Section 5], Harish-Chandra extended the concept of character for a particular class of representa-tions of a real reductive Lie group. More precisely, he proved that for a quasi-simple representation ( Π , H ) (see[7, Section 10]) of a real reductive Lie group G, the operator Π ( Ψ ) , Ψ ∈ C ∞ c (G), given by Π ( Ψ ) = Z G Ψ ( g ) Π ( g ) dg , is a trace class operator and the corresponding map Θ Π : C ∞ c (G) ∋ Ψ → tr( Π ( Ψ )) ∈ C is a distribution; Θ Π is usually called the distribution character of Π . Moreover, Harish-Chandra proved (see [9,Theorem 2]) that there exists a locally integrable function Θ Π on G, analytic on G reg (where G reg is the set ofregular points of G, see [9, Section 3]), such that Θ Π ( Ψ ) = Z G Θ Π ( g ) Ψ ( g ) dg , (cid:0) Ψ ∈ C ∞ c (G) (cid:1) . The locally integrable function Θ Π is the character of Π . In few cases, an explicit value of Θ Π is well-known:(1) G compact (H. Weyl),(2) ( Π , H ) a discrete series representation:(a) Harish-Chandra (see [8]) established a formula for Θ Π on the compact Cartan subgroup T of G,(b) Hecht (see [10]) determined the value of Θ Π on every Cartan subgroup of G for holomorphic discreteseries representation,(3) ( Π , H ) irreducible unitary highest weight module (Enright, [5, Corollary 2.3], see also [20]).The goal of this paper is to explain how to use Howe’s correspondence and the Cauchy–Harish-Chandra integralintroduced by T. Przebinda to get explicit values of characters for some particular irreducible unitary non-highestweight modules of U( n , n +
1) starting from a representation of U(1). Our method is as follows. Let (G , G ′ ) = (U(1) , U( p , q )) , p , q ≥
1, be a dual pair in Sp(2( p + q ) , R ), f Sp(2( p + q ) , R ) be the corresponding metaplectic group(see Equation (1)), ω p , q be the metaplectic representation of f Sp(2( p + q ) , R ) (see Theorem 2.2), e G and f G ′ be the Mathematics Subject Classification.
Primary: 22E45; Secondary: 22E46, 22E30.
Key words and phrases.
Howe correspondence, Characters, Cauchy–Harish-Chandra integral, Orbital Integrals. reimages of G and G ′ in f Sp(2( p + q ) , R ) and Π be a representation of U(1). We denote by θ p , q the map comingfrom Howe’s duality theorem (see Equation (3)) θ p , q : R ( e U(1) , ω p , q ) → R ( e U( p , q ) , ω p , q ) , where R ( e U(1) , ω p , q ) and R ( e U( p , q ) , ω p , q ) are defined in Notation 3.4. By assumption on p and q , Π ′ : = θ p , q ( Π )is a non-zero irreducible unitary highest weight module of e U( p , q ). In Appendix A, we computed the value of thecharacter Θ Π ′ of Π ′ on every Cartan subgroup of e U( p , q ). A similar result was obtained in [21] for p = q = ′ , G n ) = (U( p , q ) , U( n , n + p + q )(2 n + , R ). As before, we denoteby f Sp(2( p + q )(2 n + , R ) the metaplectic group of Sp(2( p + q )(2 n + , R ), ω p , qn , n + the metaplectic representationof f Sp(2( p + q )(2 n + , R ) and θ p , qn , n + : R ( e U( p , q ) , ω p , qn , n + ) → R ( e U( n , n + , ω p , qn , n + )the map obtained from Howe’s correspondence (see Equation (3)). Using a result of Kudla (see [18]), it followsthat Π n : = θ p , qn , n + ( Π ′ ) , n ≥
2, i.e. Π n ∈ R ( e U( n , n + , ω p , qn , n + ). Note that by using [19], it follows that Π n = Π n , where Π n is usually called the "big theta" and is defined in Section 3.As explained in [25] (see also Remark 4.5), by using that Π ′ is unitary, we get that if p + q ≤ n , the distributioncharacter Θ Π n of Π n can be obtained by using the Cauchy–Harish-Chandra integral (see Section 4). More precisely,because Π n = Π n , we get from Equation (4), Theorem 5.5 and Equation (8) that: Θ Π n ( Ψ ) = min( p , q ) + X i = Z e H ′ i Θ Π ′ (˜ h ′ i ) (cid:12)(cid:12)(cid:12) det(Id − Ad(˜ h ′ i ) − ) g ′ / h ′ i (cid:12)(cid:12)(cid:12) Chc ˜ h ′ i ( Ψ ) d ˜ h ′ i , (cid:16) Ψ ∈ C ∞ c ( f G n ) (cid:17) , where H ′ , . . . , H ′ min( p , q ) + is a maximal set of non-conjugate Cartan subgroups of G ′ and Chc ˜ h ′ i is a family ofdistributions on f G n parametrized by regular elements on the di ff erent Cartan subgroups of f G ′ as recalled in Section4.In this paper, we compute explicitly the value of Θ Π n on every Cartan subgroup of f G n for p = q =
1. We keepthe notations of Appendix B and parametrize the n + n by subsets S = {∅} , S , . . . , S n ofstrongly orthogonal imaginary roots of g n (see also [26, Section 2]) and let H n (S ) , . . . , H n (S n ) be the correspondingCartan subgroups of G n and H n , S , . . . , H n , S n be the diagonal subgroups of GL(2 n + , C ) given, for 0 ≤ t ≤ n , byH n , S t = c (S t ) − H n (S t ) c (S t ), where c (S t ) is the Cayley transform defined in Equation (22).In Theorem 6.5, we explain how to go from the distribution character Θ Π n to the locally integrable function Θ Π n byusing results of Bernon and Przebinda from [3] and [2] (see also Section 5). In Theorem 6.10, we computed thevalue of Θ Π n of the di ff erent Cartan subgroups of G n and get for every t ∈ [ | , n | ], Θ Π n ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) = ± C − e A P j ∈ K ( h ) ∪ A ti ∈ J ( h ) ∪ B t h ni h n + mj Ω i , j ( h ) + δ t , B e − ( m + X n + ) X n + Σ ( h ) if m ≥ e A P i , j ∈ J ( h ) ∪ B ti , j h ni h nj Ω i , j ( h ) + δ t , B e − ( m + X n + ) X n + Σ ( h ) if m = e A P i ∈ K ( h ) ∪ A tj ∈ J ( h ) ∪ B t h n + mi h nj Ω i , j ( h ) + δ t , B e ( m − X n + ) X n + Σ ( h ) if m ≤ − m is the highest weight of Π (see Notation A.1), ˇH n , S t is a double cover of H n , S t defined inEquation (5), ˇ p : ˇH n , S t → e H n , S t is defined in Section 5, h is the element of H n , S t given, as in Equation (17), by h = ( h , . . . , h n + ) = diag( e iX − X n + , . . . , e iX t − X n + − t , e iX t + , . . . , e iX n + − t , e iX t + X n + − t , . . . , e iX + X n + ) , X j ∈ R , here Ω i , j , ≤ i , j ≤ n +
1, and Σ are the functions on H reg n , S t are given by Ω i , j ( h ) = n + Q k = k , i , j h k n + Q k = k , i ( h i − h k ) n + Q l = l , i , j ( h j − h l ) , Σ ( h ) = sgn( X n + ) e imX (cid:12)(cid:12)(cid:12) e (2 n − X n + (cid:12)(cid:12)(cid:12) (cid:16) − e − X n + (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Q k = (cid:16) − h h − k (cid:17) n Q k = (cid:16) − h k h − n + (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) − e − X n + (cid:12)(cid:12)(cid:12) , K ( h ) and J ( h ) are the subsets of { , . . . , t } defined by J ( h ) = { j ∈ { , . . . , t } , sgn( X n + − j ) = } , K ( h ) = { j ∈ { , . . . , t } , sgn( X n + − j ) = − } , A t and B t are the subsets of { , . . . , n + } given byA t = { t + , . . . , n } , B t = { n + , . . . , n + − t } , sgn is the sign-function on R ∗ given bysgn( X ) = X > − X < , ( X ∈ R ∗ ) , and e A and B are constants defined in Theorems 6.5 and 6.10. As explained in Lemma 6.11, the denominator n Y k = (cid:16) − h h − k (cid:17) n Y k = (cid:16) − h k h − n + (cid:17) is real and its sign is constant on every Weyl chamber.Note that the representation Π n is irreducible and unitary but not an highest weight module. In particular, itscharacter Θ Π n cannot be obtained by using Enright’s formula (see [5, Corollary 2.3]) or [20].Our formula for the character is Weyl denominator free.The method we used in this paper gives a general procedureto give characters of non-highest weight representations by starting from an highest weight representation of acompact group. In particular, proving Conjecture 4.4 could make our method more general, by removing theassumption of stable range for the second lifting.The paper is organised as follows. In Section 2, we recalled a construction of the metaplectic representation givenby Aubert and Przebinda in [1]. The goal is to define the embedding T of the metaplectic group into the set oftempered distributions on the symplectic space W (see Equation (2)) which is crucial in the construction of theCauchy–Harish-Chandra integral. After recalling Harish-Chandra’s character theory and Howe’s correspondencein Section 3, we define in Section 4 the Cauchy–Harish-Chandra integral and explain a conjecture of Przebinda onthe transfer of characters in the theta correspondence (see Conjecture 4.4). In Section 5, we summarized the resultsof [3] and [2] on how to compute the Cauchy–Harish-Chandra integral on the di ff erent Cartan subgroups, and weadapt the results to unitary groups, which are the one we consider in this paper, and make the computations for Θ Π n in Section 6 (see Theorem 6.5 and Proposition 6.10). The document contains three appendices: in Appendix A, wemake computations for Π ′ for the dual pair (G , G ′ ) = (U(1) , U( p , q )) on every Cartan subgroup of G ′ , in AppendixB, we recall how to parametrize the Cartan subgroups of U( p , q ) by using strongly orthogonal roots (see also [26,Section 2]) and in Appendix C, we define a character ε appearing in the formulas for the Cauchy–Harish-Chandraintegrals and proved a lemma for ε useful in the proof of Lemma 6.7. Acknowledgements:
I would like to thank Tomasz Przebinda for the useful discussions during the preparationof this paper. This research was supported by the MOE-NUS AcRF Tier 1 grants R-146-000-261-114 and R-146-000-302-114. . M etaplectic R epresentation Let χ be the character of R given by χ ( t ) = e i π t and let W be a finite dimensional vector over R endowed with anon-degenerate, skew-symmetric, bilinear form h· , ·i . We denote by Sp(W) the corresponding group of isometries,i.e. Sp(W) = { g ∈ GL(W) , h g ( w ) , g ( w ) i = h w , w i , ( ∀ w , w ∈ W) } , and by sp (W) its Lie algebra given by: sp (W) = { X ∈ End(W) , h X ( w ) , w i + h w , X ( w ) i = , ( ∀ w , w ∈ W) } . We first start by recalling the construction of the metaplectic group f Sp(W): it is a connected two-fold cover ofSp(W). We use the formalism of [1]. Let J be a compatible positive complex structure on W, i.e. an element ofthe Lie algebra sp (W) satisfying J = − Id W and such that the symmetric form h J · , ·i is positive definite. For everyelement g ∈ Sp( W ), we denote by J g the element of End(W) given by J g = J − ( g − g to J g (W) is invertible and let f Sp(W) be the subset of Sp(W) × C × defined by(1) f Sp(W) = n ˜ g = ( g , ξ ) ∈ Sp(W) × C × , ξ = i dim R (W) det(J g ) − g (W) o , where det(J g ) − g (W) denotes the determinant of the endomorphism J g restricted to J g (W). On f Sp(W), we define amultiplication by; ( g , ξ )( g , ξ ) = ( g g , ξ ξ C( g , g )) (cid:0) g , g ∈ Sp(W) , ξ , ξ ∈ C × (cid:1) , where C : Sp(W) × Sp(W) → C × is a cocycle defined in [1, Proposition 4.13]. Let Θ be the map defined by: Θ : f Sp(W) ∋ ˜ g = ( g , ξ ) → ξ ∈ C × . One can check easily that f Sp(W) is a connected two-fold cover of Sp(W), where the covering map π : f Sp(W) → Sp(W) is given by π (( g , ξ )) = g .For every g ∈ End(W), we denote by c ( g ) the Cayley transform of g defined by: c ( g ) : ( g − ∋ ( g − w → ( g + w + Ker( g − ∈ W / Ker( g − . We denote by S(W) the Schwartz space of W and by S ∗ (W) the corresponding space of tempered distributions. Wedefine the map t : f Sp(W) → S ∗ (W) by t ( g ) = χ c ( g ) µ ( g − , where χ c ( g ) is the function on ( g − χ c ( g ) ( w ) : ( g − → χ h c ( g ) w , w i ! , ( w ∈ ( g − , and µ ( g − is the Lebesgue measure on ( g − h J · , ·i is 1. More precisely, t ( g ) φ = Z ( g − χ c ( g ) ( w ) φ ( w ) d µ ( g − ( w ) , ( φ ∈ S(W)) . We define the map T : f Sp(W) → S ∗ (W) given by(2) T( ˜ g ) = Θ ( ˜ g ) t ( g ) (cid:16) ˜ g ∈ f Sp(W) , g = π ( ˜ g ) (cid:17) . Remark . Let ˜ g , ˜ g ∈ f Sp(W). The question of the relation between the distributions T( ˜ g ) , T( ˜ g ) and T( ˜ g ˜ g )arises naturally. In order to explain this link, we need to recall the notion of twisted convolution.For two functions φ , φ ∈ S(W), we define φ ♮φ the function on W given by φ ♮φ ( w ) = Z W φ ( u ) φ ( w − u ) χ h u , w i ! d µ W ( u ) , ( w ∈ W) . One can easily check that φ ♮φ ∈ S(W). We extend ♮ to some tempered distributions on W. In fact, for every g ∈ Sp( W ), the twisted convolution t ( g ) ♮φ ( w ) = Z ( g − χ c ( g ) ( u ) φ ( w − u ) χ h u , w i ! d µ W ( u ) , ( w ∈ W , φ ∈ S(W)) , s still a Schwartz function and the map: S(W) ∋ φ → t ( g ) ♮φ ∈ S(W)is well-defined and continuous (see [1, Proposition 4.11]). Similarly, T( ˜ g ) ♮φ ∈ S(W) for every ˜ g ∈ f Sp(W) and φ ∈ S(W). In particular, it makes sense to consider T( ˜ g ) ♮ (cid:0) T( ˜ g ) ♮φ (cid:1) for every ˜ g , ˜ g ∈ f Sp(W) and φ ∈ S(W) andone can prove that T( ˜ g ) ♮ (T( ˜ g ) ♮φ ) = T( ˜ g ˜ g ) ♮φ .Let W = X ⊕ Y be a complete polarization of the space W and we denote by dx , dy the Lebesgue measures on Xans Y respectively such that d µ W = dxdy . Using the Weyl transform K , we have a natural isomorphism betweenthe spaces S(W) and S(X × X) given by K : S(W) ∋ φ → K ( φ )( x , x ) = Z Y φ ( x − x + y ) χ h y , x + x i ! dy ∈ S(X × X) , which extends to an isomorphism on the corresponding spaces of distributions. Similarly, every tempered distri-bution on X × X can be identified to an element of Hom(S(X) , S ∗ (X)) using the Schwartz Kernel Theorem (see [1,Equation 146]). The corresponding isomorphism will be denoted by Op and let ω : f Sp(W) → Hom(S(X) , S ∗ (X))be the map given by ω = Op ◦ K ◦ T . As proved in [1, Section 4], we get that for every ˜ g ∈ f Sp(W) and v ∈ S(X), ω ( ˜ g ) v ∈ S(X) and that ω ( ˜ g ˜ h ) = ω ( ˜ g ) ◦ ω (˜ h ) for every ˜ g , ˜ h ∈ f Sp(W). The operator ω ( ˜ g ) ∈ Hom(S(X) , S(X)) can be extended to L (X) by ω ( ˜ g ) φ = lim || φ − v || → v ∈ S(X) ω ( ˜ g ) v , (cid:16) φ ∈ L (X) (cid:17) . Theorem 2.2.
For every ˜ g ∈ e S(W) and φ ∈ L (X) , the map f Sp(W) ∋ ˜ g → ω ( ˜ g ) φ ∈ L (X) , is well-defined and continuous. Moreover, ω ( ˜ g ) ∈ U(L (X)) , i.e. ω is a faithful unitary representation of f Sp(W) ,and for every Ψ ∈ C ∞ c ( f Sp(W)) , we get: Z f Sp(W) Θ ( ˜ g ) Ψ ( ˜ g ) d ˜ g = tr Z f Sp(W) Ψ ( ˜ g ) ω ( ˜ g ) d ˜ g , where d ˜ g is a Haar measure on f Sp(W) .Remark . (1) Let Sp c (W) be the subset of Sp(W) given by Sp c (W) = (cid:8) g ∈ Sp(W) , det( g − , (cid:9) . This isthe domain of the Cayley transform. We will denote by f Sp c (W) the preimage of Sp c (W) in f Sp(W).For every g ∈ Sp c (W), c ( g ) ∈ sp (W). We denote by sp c (W) the subspace of sp (W) defined by c (Sp c (W)).Obviously, c ( g ) = g . It defines a bijective map c : sp c (W) → Sp c (W). Fix an element f − π − ( {− } ). Inparticular, there exists a unique map ˜ c : sp c (W) → f Sp c (W) such that c = π ◦ ˜ c and ˜ c (0) = f − Ψ ∈ Sp(W) whose support is included in f Sp c (W), we get: Z f Sp(W) Ψ ( ˜ g ) d ˜ g = Z sp (W) Ψ (˜ c ( X )) j sp (W) ( X ) dX , where j sp (W) ( X ) = | det(1 − X ) | r , where r = R ( sp (W))dim R (W) (see [23, Section 3]).(2) For every Ψ ∈ C ∞ c ( f Sp(W)), we can consider the following distribution on W Z f Sp(W) Ψ ( ˜ g )T( ˜ g ) d ˜ g . This distribution is in fact given by a Schwartz function. Indeed, let’s first assume that the support of Ψ inincluded in f Sp c (W). For every φ ∈ S(W), we get: Z f Sp( W ) Ψ ( ˜ g )T( ˜ g ) d ˜ g ! ( φ ) = Z f Sp( W ) Ψ ( ˜ g )T( ˜ g ) φ d ˜ g Z f Sp(W) Ψ ( ˜ g ) Z W Θ ( ˜ g ) χ c ( g ) ( w ) φ ( w ) dwd ˜ g = Z sp (W) Z W Ψ (˜ c ( X )) Θ (˜ c ( X )) χ X ( w ) j sp (W) ( X ) φ ( w ) dwdX = Z W Z sp (W) Φ Ψ ( X ) χ ( τ sp (W) ( w )( X )) dX ! φ ( w ) dw = Z W F ( Φ Ψ ) ◦ τ sp (W) ( w ) φ ( w ) dw , where Φ Ψ ( X ) = Ψ (˜ c ( X )) Θ (˜ c ( X )) j sp (W) ( X ) , X ∈ sp (W), is smooth and compactly supported function on sp (W) such that supp( Φ Ψ ) ⊆ sp c (W), F ( Φ Ψ ) is the Fourier transform of Φ Ψ and τ sp (W) : W → sp (W) ∗ isthe moment map defined by τ sp (W) ( w )( X ) = h X ( w ) , w i , w ∈ W , X ∈ sp (W). In particular, F ( Φ Ψ ) ◦ τ sp (W) isa Schwarz function on W .We can remove the assumption on the support of Ψ by using the previous result. Indeed, the Zariskitopology on Sp( W ) is noetherian. In particular, there exists g , . . . , g m ∈ Sp( W ) c such that f Sp( W ) = m M i = ˜ g i f Sp c ( W ) . We can find functions Ψ , . . . , Ψ m ∈ C ∞ c ( f Sp c ( W )) such that for every ˜ g ∈ f Sp( W ),1 = m X i = Ψ i ( ˜ g i − ˜ g ) . Then, for every Ψ ∈ C ∞ c ( f Sp( W )), we get: Z f Sp( W ) Ψ ( ˜ g )T( ˜ g ) d ˜ g = m X i = Z f Sp( W ) Ψ i ( ˜ g i − ˜ g )) Ψ ( ˜ g )T( ˜ g ) d ˜ g = m X i = Z f Sp( W ) Ψ i ( ˜ g ) Ψ ( ˜ g i ˜ g )T( ˜ g i ˜ g ) d ˜ g = m X i = T( ˜ g i ) ♮ Z f Sp( W ) Ψ i ( ˜ g ) Ψ ( ˜ g i ˜ g )T( ˜ g ) d ˜ g ! . The result follows from Remark 2.1.3. C haracter T heory and H owe ’ s correspondence Let G be a real connected reductive Lie group, g = Lie(G) its Lie algebra and g C = g ⊗ R C its complexification.We denote by U ( g C ) the enveloping algebra of g (see [17, Chapter 3.1]), Z( U ( g C )) its center and D(G) the set ofdi ff erential operators on G and by D G (G) the set of left-invariant di ff erential operators on G. As explained in [11,Chapter 2], D G (G) is isomorphic to U ( g C ). Let D GG (G) be the set of bi-invariant di ff erential operators on G (whichis isomorphic to Z( U ( g C )), see [11]), D ′ (G) be the set of distributions on G and D ′ (G) G the set of G-invariantdistributions. Definition 3.1.
We say that T ∈ D ′ (G) is an eigendistribution if there exists χ T : D GG (G) → C an homomorphismof algebras such that D( T ) = χ T (D) T for every D ∈ D GG (G).We will denote by Eig(G) the set of eigendistributions on G. Theorem 3.2 (Harish-Chandra, [6]) . For every G -invariant eigendistribution T on G , there exists a locally inte-grable function f T on G , analytic on G reg , such that T = T f T , i.e. for every function Ψ ∈ C ∞ c (G) ,T ( Ψ ) = Z G f T ( g ) Ψ ( g ) dg , where dg is a Haar measure on G . Let ( Π , H ) be an irreducible quasi-simple representation (see [7, Section 10]). As explained in [6], the map Θ Π : C ∞ c (G) ∋ Ψ → tr( Π ( Ψ )) ∈ C s well-defined and is a distribution (in the sense of Laurent Schwartz). In particular, by assumption of Π , it followsfrom Theorem 3.2 that there exists Θ Π ∈ L (G) such that Θ Π ( Ψ ) = Z G Θ Π ( g ) Ψ ( g ) dg for every Ψ ∈ C ∞ c (G). The function Θ Π is called the character of Π .We now recall Howe’s duality theorem and how it can be studied through characters. Let W be a finite dimensionalvector space over R endowed with a non-degenerate, skew-symmetric, bilinear form h· , ·i . As in Section 2, wedenote by Sp(W) the group of isometries of (W , h· , ·i ), by ( f Sp( W ) , π ) the metaplectic cover of Sp(W) (see Equation(1)), by ( ω, H ) the corresponding Weil representation (see Theorem 2.2) and by ( ω ∞ , H ∞ ) the correspondingsmooth representation (see [27, Chapter 0]).A dual pair in Sp(W) is a pair of subgroups (G , G ′ ) of Sp(W) which are mutually centralizer in Sp( W ). The dualpair is called irreducible if we cannot find any orthogonal decomposition W = W ⊕ W where both spaces W and W are G · G ′ -invariant, and called reductive if the actions of G and G ′ on W are both reductive. The set ofirreducible reductive dual pairs in Sp(W) had been classified by Howe in [14]. Remark . In this paper, we will focus our attention on a dual pair consisting of two unitary groups. Moreprecisely, let V and V ′ be two complex vector spaces endowed with an hermitian form ( · , · ) and skew-hermitianform ( · , · ) ′ respectively. We denote by U(V) and U(V ′ ) the corresponding group of isometries and by W the complexvector space given by W = V ⊗ C V ′ . The space W can naturally be seen as a real vector space, and to avoid anyconfusion, we will denote by W R the corresponding real vector space. The skew-hermitian form b = ( · , · ) ⊗ ( · , · ) ′ on W defines a skew-symmetric form h· , ·i on W R by h· , ·i = Im(b). In particular, (U(V) , U(V ′ )) is a dual pair inSp(W R , h· , ·i ).If we denote by ( p , q ) and ( r , s ) the signatures of ( · , · ) and ( · , · ) ′ respectively, we get that (U( p , q ) , U( r , s )) form adual pair in Sp(2( p + q )( r + s ) , R ). Notation 3.4.
For a subgroup H of Sp(W), we denote by e H = π − (H) the preimage of H in f Sp(W) and let R ( e H , ω )be the set of equivalence classes of irreducible admissible representations of ˜G which are infinitesimally equivalentto a quotient of H ∞ by a closed ω ∞ ( e H)-invariant subspace.
Theorem 3.5 (R. Howe, [15]) . For every reductive dual pair (G , G ′ ) of Sp(W) , we get a bijection between R ( e G , ω ) and R ( f 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, the space 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, not irreducible in general, but Howe’s duality theorem says that there exists a uniqueirreducible quotient Π ′ of Π ′ with Π ′ ∈ R ( f G ′ , ω ) and Π ⊗ Π ′ ∈ R ( e G · f G ′ , ω ).We will denote by(3) θ : R ( e G , ω ) → R ( f G ′ , ω )the corresponding bijection. Remark . If G is compact, the situation turns out to be slightly easier. The action of e G on H ∞ can be decom-posed as H ∞ = M ( Π , H Π ) ∈ R ( e G ,ω ) H ( Π ) , where H ( Π ) is the closure of n T ( H Π ) , T ∈ Hom e G ( H Π , H ∞ ) , { } o and R ( e G , ω ) is the set of representations( Π , V Π ) of e G such that Hom e G (V Π , ω ∞ ) , { } . Obviously, f G ′ acts on H ( Π ) and we get that H ( Π ) = Π ⊗ Π ′ where Π ′ is an irreducible unitary representation of f G ′ . . C auchy H arish -C handra integral and transfer of invariant eigendistributions We start this section by recalling the construction of the Cauchy–Harish-Chandra integral introduced in Section[24, Section 2]..Let (G , G ′ ) be an irreducible reductive dual pair in Sp(W) and T : f Sp(W) → S ∗ (W) the map defined in Equation(2). Let H , . . . , H n be a maximal set of non-conjugate Cartan subgroups of G and let H i = T i A i the decompositionof H i as in [28, Section 2.3.6], where T i is maximal compact in H i . For every 1 ≤ i ≤ n , we denote by A ′ i thesubgroup of Sp(W) given by A ′ i = C Sp(W) (A i ) and let A ′′ i = C Sp(W) (A ′ i ). As recalled in [24, Section 1], there existsan open and dense subset W A ′′ i , which is A ′′ i -invariant and such that A ′′ i \ W A ′′ i is a manifold, endowed with ameasure dw such that for every φ ∈ C ∞ c (W) such that supp( φ ) ⊆ W A ′′ i , Z W A ′′ i φ ( w ) dw = Z A ′′ i \ W A ′′ i Z A ′′ i φ ( aw ) dadw . For every Ψ ∈ C ∞ c ( e A ′ i ), we denote by Chc( Ψ ) the following integral:Chc( Ψ ) = Z A ′′ i \ W A ′′ i T( Ψ )( w ) dw . According to Remark 2.3, the previous integral is well-defined and as proved in [24, Lemma 2.9], the correspondingmap Chc : C ∞ c ( e A ′ i ) → C defines a distribution on e A ′ i . Remark . We say few worlds about the dual pair (A ′ i , A ′′ i ) and the space W A ′′ i . Let V , i be the subspace of V onwhich A i acts trivially and V , i = V ⊥ , i . The restriction of ( · , · ) to V , i is non-degenerate and even dimensional. Inparticular, there exists a complete polarization of V , i of the form V , i = X i ⊕ Y i , where both spaces X i and Y i areH i -invariant.By looking at the action of A i on V , i , we get:X i = X i ⊕ . . . ⊕ X ki , Y i = Y i ⊕ . . . ⊕ Y ki , where all the spaces X ji , ≤ i ≤ n , ≤ j ≤ k , are A i -invariant and mutually non-equivalent. In particular,W = Hom(V , V ′ ) = Hom(V , i , V ′ ) ⊕ Hom(V , i , V ′ ) = Hom(V , i , V ′ ) ⊕ k M j = (cid:16) Hom(X ji , V ′ ) ⊕ Hom(Y ji , V ′ ) (cid:17) . To simplify the notations, we denote by W ij the subspace of W given by Hom(X ji , V ′ ) ⊕ Hom(Y ji , V ′ ) and W , i = Hom(V , i , V ′ ). One can easily check that:A ′ i = Sp(W , i ) × GL(Hom(X i , V ′ ) R ) × . . . × GL(Hom(X ki , V ′ ) R )and A ′′ i = O(1) × GL(1 , R ) × . . . × GL(1 , R ) . Moreover, W A ′′ i = (W , i \ { } ) × e W , i × . . . × e W n , i , where e W j , i = n ( x , y ) ∈ Hom(X ji , V ′ ) ⊕ Hom(Y ji , V ′ ) , x , , y , o , 1 ≤ j ≤ k .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 . As proved in [24], for every ˜ h i ∈ e H i reg , the pull-back τ ∗ ˜ h i (Chc) of Chc via τ ˜ h i (see [13, Theorem 8.2.4]) is a well-defined distribution on f G ′ .For every ˜ h i ∈ e H i reg , we denote by Chc ˜ h i : = τ ∗ ˜ h i (Chc) the corresponding distribution on f G ′ . otation 4.2. For every reductive group G, we denote by I (G) the space of orbital integrals on G as in [4,Section 3], endowed with a natural topology defined in [4, Section 3.3]. We denote by J G the map J G : C ∞ c (G) → C ∞ (G reg ) G given as follows: for every γ ∈ G reg , there exists a unique, up to conjugation, Cartan subgroups H( γ ) ofG such that γ ∈ H( γ ), and for every Ψ ∈ C ∞ c (G), we define J G ( Ψ )( γ ) by:J G ( Ψ )( γ ) = (cid:12)(cid:12)(cid:12) det(Id − Ad( γ − )) g / h ( γ ) (cid:12)(cid:12)(cid:12) Z G / H( γ ) Ψ ( 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 let J t G : I (G) ∗ → D ′ (G) be the transpose of J G . In [4, Theorem 3.2.1], Bouaziz proved that the mapJ t G : I (G) ∗ → D ′ (G) G is bijective.We now apply these results to construct a map Chc ∗ , transferring the invariant distributions for a given dual pair(G , G ′ ). Let (G , G ′ ) be an irreducible dual pair in Sp( W ) such that rk(G) ≤ rk(G ′ ). For every function Ψ ∈ C ∞ c ( f G ′ ),we denote by g Chc( Ψ ) the e G-invariant function on e G reg given by: g Chc( Ψ )(˜ h i ) = Chc ˜ h i ( Ψ ) , (˜ h i ∈ e H i reg ) . As proved in [3], the corresponding map g Chc : C ∞ c ( f G ′ ) → I ( e G)is well-defined and continuous and factors through I ( f G ′ ), i.e. g Chc : I ( f G ′ ) → I ( e G)and the corresponding map is continuous. In particular, we get a mapChc ∗ : D ′ ( e G) e G ∋ T → J t f G ′ ◦ g Chc t ◦ (J t e G ) − ( T ) ∈ D ′ ( f G ′ ) f G ′ . Theorem 4.3.
The map
Chc ∗ sends Eig( e G) e G into Eig( f G ′ ) f G ′ . Moreover, if Θ is a distribution on e G given by a locallyintegrable function Θ on f G ′ , we get for every Ψ ∈ C ∞ c ( f G ′ ) that: (4) Chc ∗ ( Θ )( Ψ ) = n X i = | W (H i ) | Z e H i reg Θ (˜ h i ) | det(1 − Ad(˜ h − i )) g / h i | Chc( Ψ )(˜ h i ) d ˜ h i , where H , . . . , H n is a maximal set of non-conjugate Cartan subgroups of G . In [24], T. Przebinda conjectured that the correspondence of characters in the theta correspondence should beobtained via Chc ∗ . More precisely, Conjecture 4.4.
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 ′ . Remark . The conjecture is known to be true in few cases:(1) G compact,(2) (G , G ′ ) in the stable range and Π a unitary representation of e G (see [25]),(3) (G , G ′ ) = (U( p , q ) , U( r , s )), with p + q = r + s and Π a discrete series representation of e G (see [22]). . E xplicit formulas of Chc for unitary groups
In this section, we quickly explain how to compute explicitly the Cauchy–Harish-Chandra integral on the dif-ferent Cartan subgroups. Because our papers will only concerns characters of some unitary groups, we will adaptthe results of [3] and [2] in this context, but similar results can be obtained for other dual pairs.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 = { f , . . . , f n } , n = p + q (resp. B V ′ = n f ′ , . . . , f ′ n ′ o , n ′ = r + s )be a basis of V (resp. V ′ ) such that Mat (( · , · ) , B V ) = Id p , q (resp. Mat (cid:0) ( · , · ) ′ , B V ′ (cid:1) = i Id r , s ). Let G and G ′ be thecorresponding 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 K = U( p ) × U( q ) and K ′ = U( r ) × U( s ) be the maximal compact subgroups of G and G ′ respectively and let Hand H ′ be the diagonal Cartan subgroups of K and K ′ respectively. By looking at the action of H on 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 i f 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 i f ′ 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 byi w = J ◦ w , ( w ∈ W) . We 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 ′ . Notation 5.1.
We denote by ∆ (resp. ∆ ( k )) the root system corresponding to ( g C , h C ) (resp. ( k C , h C )), by Ψ (resp. Ψ ( k )) a system of positive roots of ∆ (resp. ∆ ( k )) and let Φ = − Ψ (resp. Φ ( k ) = − Ψ ( k )) the set of negative roots.Let e i be the linear form on h C = C p + q given by e i ( λ , . . . , λ p + q ) = λ i . As explained in [17, Chapter 2], we knowthat: ∆ = n ± ( e i − e j ) , ≤ i < j ≤ p + q o , Ψ = n e i − e j , ≤ i < j ≤ p + q o , and ∆ ( k ) = n ± ( e i − e j ) , ≤ i < j ≤ p o ∪ n ± ( e i − e j ) , p + ≤ i < j ≤ p + q o , Ψ ( k ) = Ψ ∩ ∆ ( k ) . We define ∆ ′ , ∆ ′ ( k ) , Ψ ′ , Ψ ′ ( k ) , Φ ′ , Φ ′ ( k ) similarly and denote by e ′ i , ≤ i ≤ r + s the linear form on h ′ C = C r + s givenby e i ( λ , . . . , λ r + s ) = λ i .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 . e denote by ˇH ′ C the connected two-fold cover of H ′ C isomorphic to(5) h ′ C / n ′ X j = π x j J ′ j , n ′ X j = x j ∈ Z , x j ∈ Z . One can easily check that ρ ′ = P α ∈ Ψ ′ α is analytic integral on ˇH ′ C . As explained in [24, Section 2], we construct amap ˇ p : ˇH ′ C → e H ′ C which is bijective (but not an isomorphism of covering of H ′ C in general).As explained in Appendix B, every Cartan subgroup of G ′ can be parametrized by a subset S ⊆ Ψ ′ st n consistingof non-compact strongly otrhogonal roots. We denote by H ′ (S) the corresponding Cartan subgroup and by H ′ S the subgroup of H ′ C as in Appendix B. Let S ⊆ Ψ ′ st n and ˇH ′ S the preimage of the Cartan subgroup H ′ S in ˇH ′ C (seeAppendix B). For every ϕ ∈ C ∞ c ( f G ′ ), we denote by H S ϕ the function 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 ) , 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 ) , and by ∆ Φ ′ the function on ˇH ′ regS given by ∆ Φ ′ (ˇ h ′ ) = ˇ h ′ P α ∈ Φ ′ α Q α ∈ Φ ′ (1 − ˇ h ′− α ). Remark . For every ˇ h ′ ∈ ˇH ′ regS , ∆ Φ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ) = Q α ∈ Ψ ′ + (1 − ˇ h ′ α )(1 − ˇ h ′− α ) = Q α ∈ Ψ ′ + (1 − ˇ h ′ α )(1 − ˇ h ′ α ). We denote by | ∆ G ′ (ˇ h ′ ) | = ∆ Φ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ). Proposition 5.3 (Weyl’s Integration Formula) . For every ϕ ∈ C ∞ c ( f G ′ ) , we get: (6) 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 B.2).Proof. See [3, Section 2, Page 3830]. (cid:3)
Remark . One can easily see that for every S ⊆ Ψ ′ st n and ϕ ∈ C ∞ c ( f G ′ ) such that supp( ϕ ) ⊆ f G ′ · f H ′ (S) reg , theEquation (6) 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 ′ . 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 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 − { } . heorem 5.5. 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 ) | , α ∈ { , − } depends only on the choice of the positive roots Ψ and Ψ ′ , k ∈ { , − } isdefined by k = − if n ′ − n ∈ Z otherwiseand W h is the set of elements of W commuting with h . The theorem 5.5 tells us how to compute Chc ˜ h for an element ˜ h in the compact Cartan e H = e 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. Remark . One can easily check that the space W h is given byW h = n X i = Hom(V ′ i , V i ) . From now on, we assume that p ≤ q , r ≤ s and p ≤ r . Notation 5.7.
For every t ∈ [ | , p | ], we denote by S t and S ′ t the subsets of Ψ st n and Ψ ′ st n respectively given byS t = n e − e p + , . . . , e t − e p + t o , S ′ t = (cid:8) e ′ − e ′ r + , . . . , e ′ t − e ′ r + t (cid:9) , where the linear forms e k , e ′ h have been introduced in Notation 5.1.For every t ∈ [ | , p | ], we denote by H(S t ) and H ′ (S t ) the Cartan subgroups of G and G ′ respectively and let H(S t ) = T(S t )A(S t ) (resp. H ′ (S t ) = T ′ (S t )A ′ (S t )) be the decompositions of H(S t ) (resp. H ′ (S t )) as in [28, Section 2.3.6].As in Remark 4.1, we denote by V , t the subspace of V on which A(S t ) acts trivially, by V , t the orthogonalcomplement of V , t in V and by V , t = X t ⊕ Y t a complete polarization of V , t . Because p ≤ r , we have a naturalembedding of V , t into V ′ such that X t ⊕ Y t is a complete polarization with respect to ( · , · ) ′ . We denote by U t theorthogonal complement of V , t in V ′ ; in particular, we get a natural embedding:GL(X t ) × G(U t ) ⊆ G ′ = U( r , s ) . We denote by T (S t ) the maximal subgroup of T(S t ) which acts trivially on V , t and let T (S t ) the subgroup ofT(S t ) such that T(S t ) = T (S t ) × T (S t ) with T (S t ) ⊆ G(V , t ). In particular,(7) H(S t ) = T (S t ) × A(S t ) × T (S t ) . Similarly, we get a decomposition of H ′ (S ′ t ) of the form:H ′ (S ′ t ) = T ′ (S ′ t ) × A ′ (S ′ t ) × T ′ (S ′ t ) . Remark . One can easily see that for every 0 ≤ j < i ≤ r , we get:H ′ (S ′ i ) = T ′ (S ′ j ) × A ′ (S ′ j ) × H ′ ( e S ′ i − j ) , where e S ′ i − j = S ′ i \ S ′ j and H ′ ( e S ′ i − j ) is the Cartan subgroup of U( r − j , s − j ) whose split part has dimension i − j . Inparticular, H ′ S ′ i = T ′ , S ′ j × A ′ S ′ j × H ′ e S ′ i − j . et η (S t ) and η ′ (S ′ t ) be the nilpotent Lie subalgebras of u ( p , q ) and u ( r , s ) respectively given by η (S ′ t ) = Hom(X t , V , t ) ⊕ Hom(X t , Y t ) ∩ u ( p , q ) , η ′ (S ′ t ) = Hom(U t , X t ) ⊕ Hom(X t , Y t ) ∩ u ( r , s ) . We will denote by W , t the subspace of W defined by Hom(U t , V , t ) and by P(S t ) and P ′ (S ′ t ) the parabolic subgroupsof G and G ′ respectively whose Levi factors L(S t ) and L ′ (S ′ t ) are given byL(S t ) = GL(X t ) × G(V , t ) , L ′ (S ′ t ) = GL(X t ) × G(U t ) , and by N(S t ) : = exp( η (S t )) and N ′ (S ′ t ) : = exp( η ′ (S ′ t )) the unipotent radicals of P(S t ) and P ′ (S ′ t ) respectively. Remark . One can easily check that the forms on V , t and U t have signature ( p − t , q − t ) and ( r − t , s − t )respectively.As proved in [2, Theorem 0.9], for every ˜ h = ˜ t ˜ a ˜ t ∈ e H(S t ) reg (using the decomposition of H(S t ) given in Equation(7)) and ϕ ∈ C ∞ c ( f G ′ ), we get:(8) | det(Ad(˜ h ) − Id) η (S t ) | Chc ˜ h ( ϕ ) = C t d S t (˜ h ) ε (˜ t ˜ a ) Z GL(X t ) / T (S t ) × A(S t ) Z e G(U t ) ε (˜ t ˜ a ˜ y )Chc W , t (˜ t ˜ y )d ′ S t ( g ˜ t ˜ ag − ˜ y ) ϕ f K ′ f N ′ (S t ) ( g ˜ t ˜ ag − ˜ y ) d ˜ ydg , where C t is the constant defined given by(9) C t = t (2( p + q + r + s ) − t + ( r + s ) t r + s − tr + s p + q − t µ (K ′ ∩ L ′ (S ′ t ))2 t ( r + s − t ) ,ε is the character defined in [2, Lemma 6.3], d S t : e L(S t ) → R and d ′ S ′ t : e L ′ (S ′ t ) → R are given byd S t (˜ l ) = | det(Ad(˜ l ) η (S t ) ) | , d ′ S ′ t (˜ l ′ ) = | det(Ad(˜ l ′ ) η ′ (S t ) ) | , (cid:16) ˜ l ∈ e L(S t ) , ˜ l ′ ∈ e L ′ (S ′ t ) (cid:17) , and ϕ f K ′ f N ′ (S ′ t ) is the Harish-Chandra transform of ϕ , i.e. the function on e L ′ (S ′ t ) defined by: ϕ f K ′ f N ′ (S ′ t ) (˜ l ′ ) = Z f N ′ (S ′ t ) Z f K ′ ϕ (˜ k ˜ l ′ ˜ n ˜ k − ) d ˜ kd ˜ n , (cid:16) ˜ l ′ ∈ e L ′ (S ′ t ) (cid:17) . Let’s explain the method we will use in the next section to get character formulas of representations of U( n , n + , G ′ ) = (U( p ) , U( r , s )) be a dual pair in Sp(2 p ( r + s ) , R ). To avoid any confusions, we will denote by ω pr , s the metaplectic representation of Sp(2 p ( r + s ) , R ) and by θ pr , s : R ( e U( p ) , ω pr , s ) → R ( e U( r , s ) , ω pr , s ) the map defined inEquation (3).Let Π ∈ R ( e U( p ) , ω pr , s ). In this case, as explained in Remark 3.6, we get that Π ′ = Π ′ and the correspondingrepresentation Π ′ is an irreducible unitary representation of e U( r , s ) (and Π ′ ∈ R ( e U( r , s )) , ω pr , s ). In particular,Theorem 5.5 tell us how to compute Θ Π ′ on every Cartan subgroups of e U( r , s ) (see Appendix A for p = n ≥
0, we denote by G n the unitary group corresponding to an hermitian form of signature ( n , n + p ),i.e. G n = U( n , p + n ), by ω r , sn , n + p the metaplectic representation of f Sp(W n ), where W n = ( C r + s ⊗ C C n + p ) R and by θ r , sn , n + p : R ( e U( r , s ) , ω r , sn , n + p ) → R ( e U( n , n + p ) , ω r , sn , n + p ) the map as in (3).Using Kudla’s persistence principle (see [18]), we know that the representation Π ′ ∈ R ( e U( r , s ) , ω pr , s ) satisfies θ n , p + nr , s ( Π ′ ) , { } , i.e. Π ′ ∈ R ( e U( r , s ) , ω r , sn , n + p ). We denote by Π n the corresponding representation of f G n as inSection 3, by Π n ∈ R ( e U( n , n + p ) , ω r , sn , n + p ) it’s unique irreducible quotient, and by Θ Π n and Θ Π n the characters of Π n and Π n respectively.Using Theorems 3.2 and 4.3, we know that the f G n -invariant eigendistribution Θ ′ n , Π ′ : = Chc ∗ ( Θ Π ′ ) is given by alocally integrable function Θ ′ n , Π ′ on f G n , analytic on f G n reg . Note that an explicit value of Θ ′ n , Π ′ on every Cartansubgroups of f G n can be obtained using Equation (4), Theorem 5.5 and Equation (8). ccording to [19], if n ≥ r + s , we get that Π n = Π n because Π ′ is unitary and by Remark 4.5, it follows that Θ ′ n , Π ′ = Θ Π n . It is also well-known that for n ≥
1, the representations Π n are unitary but are not highest weightmodules, and in particular, it’s character cannot be obtained via Enright’s formula ([5, Corollary 2.3]).In the next section, we are going to make Θ Π n explicit for p = r = s = haracter formulas for some representations of U( n , n + , G ′ ) = (U(1) , U(1 , , R ). Because the set of irreducible genuinerepresentations of e U(1) is isomorphic to Z , the corresponding representation of R ( e U(1) , ω , ) will be denoted by Π m , m ∈ Z and let Π ′ m be the corresponding representation of f G ′ . Moreover, as explained in Section 5, Π n = θ , n , n + ( Π ′ m ) the lift of Π ′ m on f G n is non-zero and its character Θ Π n is equal to Chc ∗ ( Θ Π ′ m ). In this section, we aregoing to give an explicit formula for Θ Π n on every Cartan subgroups of f G n . Remark . We denote by g , g ′ and g n the Lie algebras of G , G ′ and G n respectively. The Lie algebra g ′ is givenby g ′ = ( a b ¯ b d ! , a , d ∈ i R , b ∈ C ) = R i i ! ⊕ R i − i ! ⊕ R ! ⊕ R i − i ! . and the two Cartan subgroups of G ′ , up to conjugation, are of the form (10)H ′ = H ′ (S ′ ) = (cid:8) diag( h , h ) , h , h ∈ U(1) (cid:9) , H ′ (S ′ ) = exp R i i ! ⊕ R !! = ( e i θ ch( X ) sh( X )sh( X ) e i θ ch( X ) ! , θ, X ∈ R ) , where S ′ = {∅} and S ′ = { e − e } (see Appendix B). We denote by (V ′ , ( · , · )) the skew-hermitian form correspond-ing to G ′ and by (V n , ( · , · ) n ) the hermitian form corresponding to G n . Let B V ′ = { f ′ , f ′ } be a basis of V ′ such thatMat B ′ ( · , · ) ′ = i Id , .We have the following complete polarization of V ′ V ′ = X ′ ⊕ Y ′ , X ′ = C ( f ′ + f ′ ) , Y ′ = C ( f ′ − f ′ ) , where both X ′ and Y ′ are H ′ -invariant. Let B V n = { f n , . . . , f n n + } be a basis of V n such that Mat B V n ( · , · ) n = Id n , n + .We consider the embedding of V ′ onto V n sending f ′ onto f n and f ′ onto f n n + . Obviously, X ′ ⊕ Y ′ is a completepolarization of V ′ ⊆ V n with respect to ( · , · ) n . We consider the subspace U of V n given byV n = V ′ ⊕ U , U = V ′⊥ , where V ′⊥ is the orthogonal complement of V ′ in V n with respect to ( · , · ) n .Let G(U ) be the group of isometries corresponding to the hermitian space (U , ( · , · ) n | U1 ). Note that G(U ) ≈ U( n − , n ).As explained in Appendix B, for every 0 ≤ t ≤ n and S t = { e − e n + , . . . , e t − e n + − t } , we denote by H n (S t )the Cartan subgroup of G n whose split part is of dimension t and by H n , S t the subgroup of H n ( ∅ ) C = { h = diag( h , . . . , h n + ) , h i ∈ C } given by H n , S t = c (S t )H n (S t ) c (S t ) − , where c (S t ) is the Cayley transform correspondingto S t (see Appendix B) We denote by P n (S ) the parabolic subgroup of G n whose Levi factor L n (S ) is given byL n (S ) = GL(X ) × G(U ). Lemma 6.2.
We get
GL(X ′ ) = H ′ (S ′ ) .Proof. The Lie algebra of GL(X ′ ) is the set of matrices A = a bc d ! of g ′ such that:(11) A ! = αα ! A − ! = β − β ! , ( α, β ∈ C ) . We first assume that A ∈ End(V ′ ) satisfies the conditions of Equation (11). Then, we get: a + b = α c + d = α a − b = β c − d = − β n particular, a + b = c + d and a − b = − c + d . Then, a = d and b = c . In particular, if A ∈ g ′ , we get that a ∈ i R and b ∈ R . In particular, GL(X ′ ) = exp R i i ! ⊕ R !! = exp( h ′ (S ′ )) = H ′ (S ′ ) . (cid:3) In this section, we are going to determine the value of the character Θ Π n on the n + ff erent Cartan subgroups ofG n . Notation 6.3.
We denote by ∆ n the set of roots corresponding to ( g n , h n ), where H n = H n ( ∅ ) is the compact Cartanof G n , by Ψ n a set of positive roots of ∆ n , by Φ n = − Ψ n , by Ψ st n ( n ) = { e t − e n + − b , ≤ b ≤ n } the corresponding setof strongly orthogonal roots of Ψ n and by Z n the subgroup of G n defined by Z n = G h ′ n , where h ′ = Lie(H ′ ) is theLie algebra of H ′ seen as a subspace of h n .We denote by η ′ (S ′ ) the subspace of g ′ defined by Hom(X ′ , Y ′ ) ∩ g ′ and by η n (S ) the subspace of g n = Lie(G n )given by Hom(X ′ , U ) ⊕ Hom(X ′ , Y ′ ) ∩ g n . Remark . As explained in Remark 5.8, we get for every t ≥ n (S t ) = T (S ) × A(S ) × H n − ( e S t − ) , where e S t − = S t \ { e − e n + } and H n − ( e S t − ) is a Cartan subgroup of G(U ) whose split part is of dimension t − h ∈ ˇH n , S t can be written as ˇ h = ˇ t ˇ a ˇ h (where, by convention, t = a = Id and ˇ h = ˇ h if t = Theorem 6.5.
For every t ∈ [ | , n | ] and ˇ h = ˇ t ˇ a ˇ h ∈ ˇH n , S t as in Remark 6.4, we get, up to a constant, that: Θ Π n ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) = A X σ ∈ W (H C n ) ε ( σ ) ∆ Φ (Z n ) ( σ − (ˇ h )) ∆ Φ n (ˇ h ) lim r ∈ E σ, S tr → Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ Ψ ′ (ˇ h ′ )det(1 − p (ˇ h ′ ) rp (ˇ h )) σ W h d ˇ h ′ + δ t , B Θ Π ′ m ( c (S ) ˇ p (ˇ t ˇ a ) c (S ) − ) | ∆ Ψ ′ (ˇ t ˇ a ) | d S ′ ( c (S ) ˇ p (ˇ t ˇ a ) c (S ) − )d ′ S ( c (S t ) ˇ p (ˇ t ˇ a ˇ h ) c (S t ) − )D( c (S t ) ˇ p (ˇ h ) c (S t ) − ) | ∆ G(U ) (ˇ h ) | ε ( g ( − c ( e S t − ) ˇ p (ˇ h ) c ( e S t − ) − ) − | det(Ad( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − ) − − | η ′ (S1) | D ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) | ∆ G n (ˇ h ) | , where δ t , = if t = otherwise , D and D are functions on e H n (S t ) , ≤ t ≤ n, given by D (˜ h ) = | det(Id − Ad(˜ h ) − ) l n (S ) / h n (S ) | , D(˜ h ) = | det(Id − Ad(˜ h ) − ) g n / h n (S ) | , (˜ h ∈ e H n (S t )) . and where A and B are constants given by A = ( − u n − , B = n + (2 n − m e S t − n + m S t . Before proving Theorem 6.5, we recall a lemma concerning orbital integrals.
Lemma 6.6.
For every t ≥ , ˜ h ∈ e H n (S t ) reg and Ψ ∈ C ∞ c ( f G n ) , we get: Z G n / H n (S t ) Ψ ( g ˜ hg − ) dg = C n , D (˜ h )D(˜ h ) Z L (S ) / H n (S t ) Ψ f K n f N n (S ) ( m ˜ hm − ) dm , where K n is the maximal compact subgroup of G n , N n (S ) = exp( η n (S )) and C n , is the constant given by C n , = µ (K n ∩ L n (S )) √ dim R ( η n (S )) = µ (K n ∩ L n (S )) √ n − . Proof.
We see easily that for t ≥
1, H n (S t ) is a Cartan subgroup of L n (S ), and then the result follows from [2,Corollary A.4]. (cid:3) roof of Theorem 6.5. Fix t ∈ [ | , n | ] and Ψ ∈ C ∞ c ( f G n ) such that supp( Ψ ) ⊆ f G n · f H n (S t ). Using Remark 5.4, itfollows that: Θ Π n ( Ψ ) = Z f G n Θ Π n ( ˜ g ) Ψ ( ˜ g ) d ˜ g = m S t Z ˇH n , S t Θ Π n ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) ε Ψ n , S t , R (ˇ h ) ∆ Φ ( n ) (ˇ h ) H S t Ψ (ˇ h ) d ˇ h = m S t Z ˇH n , S t Θ Π n ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) | ∆ G n (ˇ h ) | Z G n / H n (S t ) Ψ ( gc (S t ) ˇ p (ˇ h ) c (S t ) − g − ) dgd ˇ h . (12)According to Remark 4.5, the global character Θ Π n of Π n is given by Θ Π n ( Ψ ) = Z f H ′ Θ Π ′ m (˜ h ′ ) | det(Id − Ad(˜ h ′ ) − ) g ′ / h ′ | Chc ˜ h ′ ( Ψ ) d ˜ h ′ + Z f H ′ (S ′ ) Θ Π ′ m (˜ h ′ ) | det(Id − Ad(˜ h ′ ) − ) g ′ / h ′ (S ′ ) | Chc ˜ h ′ ( Ψ ) d ˜ h ′ , where H ′ , H ′ (S ) are the two Cartan subgroups of G ′ (up to conjugation) defined in Equation (10). Using thatsupp( Ψ ) ⊆ f G n · f H n (S t ), we get from Theorem 5.5 and [3, Equation 8] that: Z f H ′ Θ Π ′ m (˜ h ′ ) | det(Id − Ad(˜ h ′ ) − ) g ′ / h ′ | Chc ˜ h ′ ( Ψ ) d ˜ h ′ = Z f H ′ Z f G n Θ Π ′ m (˜ h ′ ) | det(Id − Ad(˜ h ′ ) − ) g ′ / h ′ | Θ ( g ( − g ˜ h ′ ) Ψ (˜ g ) d ˜ gd ˜ h ′ = Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ Ψ ′ (ˇ h ′ ) ∆ Ψ ′ (ˇ h ′ ) Z f G n Θ ( ˇ p (ˇ h ′ )˜ g ) Ψ (˜ g ) d ˜ g ! d ˇ h ′ = X σ ∈ W (H C n ) M S t ( σ ) lim r ∈ E σ, S tr → Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ Ψ ′ (ˇ h ′ ) Z ˇH reg n , S t ∆ Ψ n (ˇ h ) ∆ Φ (Z n ) ( σ − (ˇ h ))det(1 − p (ˇ h ′ ) rp (ˇ h )) σ W h ′ Z G n / H n (S t ) Ψ ( gc (S t ) ˇ p (ˇ h ) c (S t ) − g − ) dgd ˇ h ′ . (13) Similarly, by using Equation (8), we get: (14) Z f H ′ (S ′ ) Θ Π ′ m (˜ h ′ ) | det(Id − Ad(˜ h ′ ) − ) g ′ / h ′ (S ′ ) | Chc ˜ h ′ ( Ψ ) d ˜ h ′ = = ∅ C Z f H ′ (S ′ ) Θ Π ′ m (˜ h ′ ) | det(Id − Ad(˜ h ′ ) − ) g ′ / h ′ (S ′ ) | | det(Ad(˜ h ′ )) | η ′ (S ′ | | det(Ad(˜ h ′ ) − | η ′ (S ′ | Z e G(U ) | det(˜ h ′ ˜ u ) | η n (S1) | ε ( g ( − u ) Ψ e K n e N n (S ) (˜ h ′ ˜ u ) d ˜ ud ˜ h ′ otherwise where the constant C defined in Equation (9) is given byC = n + (2 n − n + C n , . and C n , is defined in Lemma 6.6. In particular, the theorem follows for t =
0, i.e. S = {∅} . From now on, weassume that t ≥
1, i.e. without loss of generality that e − e n + ∈ S t . In this case, using Remark 5.8, we get:H n (S t ) = T (S ) × A(S ) × H n − ( e S t − ) . The Cartan subgroup H n (S t ) is included in the Levi L n (S ) = GL(X ′ ) × G(U ) of P n (S ). In particular, usingLemma 6.6, Equation 13 can be written as: X σ ∈ W (H C n ) M S t ( σ )C n , lim r ∈ E σ, S tr → Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ Ψ ′ (ˇ h ′ ) Z ˇH reg n , S t ∆ Ψ n (ˇ h ) ∆ Φ (Z n ) ( σ − (ˇ h ))det(1 − p (ˇ h ′ ) rp (ˇ h )) σ W h D ( c (S t ) ˇ p (ˇ h ) c (S t ) − )D( c (S t ) ˇ p (ˇ h ) c (S t ) − ) Z L n (S ) / H n (S i ) Ψ e K n e N n (S ) ( gc (S t ) ˇ p (ˇ h ) c (S t ) − g − ) dgd ˇ hd ˇ h ′ . Similarly, Equation (14) is equal to (15)C m e S t − Z ˇT ′ , S1 Z ˇA ′ S1 Θ Π ′ m ( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − ) | det(Id − Ad( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − ) − ) g ′ / h ′ (S ) | | det(Ad( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − )) | η ′ (S1) | | det(Ad( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − ) − Id) | η ′ (S1) | Z ˇH n − , e S t − Z G(U ) / H n − ( e S t − ) | ∆ G(U ) (ˇ h ) | det( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − gc ( e S t − ) ˇ p (ˇ h ) c ( e S t − ) − g − ) | η n (S1) | ε ( g ( − c ( e S t − ) ˇ p (ˇ h ) c ( e S t − ) − ) f K n e N n (S ) ( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − gc ( e S t − ) ˇ p (ˇ h ) c ( e S t − ) − g − ) dgd ˇ hd ˇ a ′ d ˇ t ′ . where e S t − = S t \ { e − e n + } . From (12), we get: (16) Θ Π n ( Ψ ) = m S t C n , Z ˇH ′ n , S t Θ Π n ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) | ∆ G n (ˇ h ) | D ( c (S t ) ˇ p (ˇ h ) c (S t ) − )D( c (S t ) ˇ p (ˇ h ) c (S t ) − ) Z L n (S ) / H n (S t ) Ψ ( gc (S t ) ˇ p (ˇ h ) c (S t ) − g − ) dgd ˇ h and using that:L n (S ) / H n (S t ) = GL(X ) / (T (S ) × A(S )) × G(U ) / H n − ( e S t − ) = G(U ) / H n − ( e S t − ) , it follows from Equations (13), (15) and (16) that for every ˇ h = ˇ t ˇ a ˇ h ∈ ˇH n , S t = T , S i × A S i × H e S t − , m S t C n , Θ Π n ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) | ∆ G n (ˇ h ) | D ( c (S i ) ˇ p (ˇ h ) c (S t ) − )D( c (S t ) ˇ p (ˇ h ) c (S t ) − ) = C n , ( − u m S t n − ∆ Ψ n (ˇ h )D ( c (S t ) ˇ p (ˇ h ) c (S t ) − )D( c (S t ) ˇ p (ˇ h ) c (S t ) − ) X σ ∈ W (H C n ) ε ( σ ) ∆ Φ (Z n ) ( σ − (ˇ h )) lim r ∈ E σ, S tr → Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ Ψ ′ (ˇ h ′ )det(1 − p (ˇ h ′ ) rp (ˇ h )) σ W h d ˇ h ′ + m e S t − C Θ Π ′ m ( c (S ) ˇ p (ˇ t ˇ a ) c (S ) − ) | ∆ Ψ ′ (ˇ t ˇ a ) | d S ′ ( c (S ) ˇ p (ˇ t ˇ a ) c (S ) − )d ′ S ( c (S t ) ˇ p (ˇ t ˇ a ˇ h ) c (S t ) − ) ε ( g ( − c ( e S t − ) ˇ p (ˇ h ) c ( e S t − ))) | ∆ G(U ) (ˇ h ) | | det(Ad( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − ) − Id) | η ′ (S1) | and the result follows. (cid:3) Lemma 6.7.
For every ˜ g ∈ f G n , ε ( ˜ g ) = ± .Proof. The space X ′ ⊗ C V n ⊕ Y ′ ⊗ C V n is a complete polarization of V ′ ⊗ C V n . In particular, f X ′ = (X ′ ⊗ V n ) R is amaximal isotropic subspace of W = (V ′ ⊗ C V n ) R .According to Equation 24, the character ε is defined on GL( f X ′ ) c by the following formula: ε ( ˜ g ) = Θ ( ˜ g ) | Θ ( ˜ g ) | (cid:16) g ∈ GL( f X ′ ) c (cid:17) . In particular, using Equation 25, for every ˜ g ∈ e G cn , we get: ε ( ˜ g ) = det f X ′ ( ˜ g ) − | det f X ′ ( ˜ g ) − | = ± | det X ′ ( g ) | − | det X ′ ( g ) | − . Using the fact that | det X ′ ( g ) | =
1, it follows that ε ( ˜ g ) = ± (cid:3) As explained in Appendix B (see Equation (23)), for every 0 ≤ t ≤ n and S t = { e − e n + − t , . . . , e t − e n + − t } ,(17) H n , S t = n h = diag( e iX − X n + , . . . , e iX t − X n + − t , e iX t + , . . . , e iX n + − t , e iX t + X n + − t , . . . , e iX + X n + ) , X j ∈ R o . In particular, using Remark 6.4, we get that h can be written as h = tah , where t = diag( e iX , , . . . , | {z } n − , e iX ) , a = diag( e − X n + , , . . . , | {z } n − , e X n + )and h = diag(1 , e iX − X n , . . . , e iX t − X n + − t , e iX t + , . . . , e iX n + − t , e iX t + X n + − t , . . . , e iX + X n , . We get Φ (Z n ) = n e i − e j , ≤ i < j ≤ n o . In particular, for every σ ∈ S n + and ˇ h ∈ ˇH n (S t ), we get: ∆ Φ (Z n ) ( σ − (ˇ h )) ∆ Φ ( n ) (ˇ h ) = Q ≤ i < j ≤ n (cid:18) h σ ( j ) h − σ ( i ) − h − σ ( j ) h σ ( i ) (cid:19)Q ≤ i < j ≤ n + (cid:18) h i h − j − h − i h j (cid:19) = ε ( σ ) h n σ (1) h n σ (2 n +
1) 2 n + Q i = i , σ (1) ,σ (2 n + h i n + Q j = j , σ (1) (cid:16) h σ (1) − h j (cid:17) n + Q j = j , σ (1) ,σ (2 n + (cid:16) h σ (2 n + − h j (cid:17) . p to a sign, the quotient ∆ Φ (Z n ) ( σ − (ˇ h )) ∆ Φ ( n ) (ˇ h ) is "uniquely" determined by σ (1) and σ (2 n + σ ∈ S n + and i , j ∈ { , . . . , n + } such that σ (1) = i , σ (2 n + = j , we denote by ∆ ( i , j , ˇ h ) the following quantity: ∆ ( i , j , ˇ h ) = ε ( σ ) ∆ Φ (Z n ) ( σ − (ˇ h )) ∆ Φ ( n ) (ˇ h ) = h ni h nj n + Q k = k , i , j h k n + Q k = k , i ( h i − h k ) n + Q l = l , i , j (cid:16) h j − h l (cid:17) . Lemma 6.8.
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 We denote by C the constant ( − u ( p + q − . Notation 6.9.
For an element h ∈ H reg n , S t as in Equation (17), we denote by J( h ) and K( h ) the subsets of { , . . . , t } given by J ( h ) = { j ∈ { , . . . , t } , sgn( X n + − j ) = } , K ( h ) = { j ∈ { , . . . , t } , sgn( X n + − j ) = − } , where sgn( X ) is defined for every X ∈ R ∗ bysgn( X ) = X > − X < . To simplify the notations, we will denote by ( h , . . . , h n + ) the components of h .Finally, we denote by A t and B t the subsets of { , . . . , n + } given byA t = { t + , . . . , n } , B t = { n + , . . . , n + − t } . Theorem 6.10.
For every ≤ t ≤ n, the value of Θ Π n on e H n (S t ) is given by (18) Θ Π n ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) = ± C − e A P j ∈ K ( h ) ∪ A ti ∈ J ( h ) ∪ B t h ni h n + mj Ω i , j ( h ) + δ t , B e − ( m + X n + ) X n + Σ ( h ) if m ≥ e A P i , j ∈ J ( h ) ∪ B ti , j h ni h nj Ω i , j ( h ) + δ t , B e − ( m + X n + ) X n + Σ ( h ) if m = e A P i ∈ K ( h ) ∪ A tj ∈ J ( h ) ∪ B t h n + mi h nj Ω i , j ( h ) + δ t , B e ( m − X n + ) X n + Σ ( h ) if m ≤ − where Ω i , j , ≤ i , j ≤ n + , and Σ are the functions on H reg n , S t are given by Ω i , j ( h ) = n + Q k = k , i , j h k n + Q k = k , i ( h i − h k ) n + Q l = l , i , j ( h j − h l ) , Σ ( h ) = sgn( X n + ) e imX (cid:12)(cid:12)(cid:12) e (2 n − X n + (cid:12)(cid:12)(cid:12) (cid:16) − e − X n + (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Q k = (cid:16) − h h − k (cid:17) n Q k = (cid:16) − h k h − n + (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) − e − X n + (cid:12)(cid:12)(cid:12) , and where ˇ h ∈ ˇH n , S t is as in Equation (17) , e A = A(2 n − and C is a constant. roof. Let h = tah ∈ H n , S t . We denote by ( h , . . . , h n + ) be the components of h . In particular, h c = e iX c − X n + − c if 1 ≤ c ≤ te iX c if t + ≤ c ≤ n + − te iX n + − c + X c if 2 n + − t ≤ c ≤ n + . We denote by ∆ n ( l ) : = ∆ n ( l n (S )) the set of roots of l n (S ) and let Ψ n ( l ) : = ∆ n ( l ) ∩ Ψ n . One can easily check that Ψ n ( l ) = n e i − e j , ≤ i < j ≤ n o . Similarly, let Ψ n ( η n (S )) = { e − e k , ≤ k ≤ n + } ∪ { e k − e n + , ≤ k ≤ n } the roots of η n (S ). Then,D( c (S t ) ˇ p (ˇ h ) c (S t ) − )D ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) = (cid:12)(cid:12)(cid:12) det(Id − Ad( c (S t ) ˇ p (ˇ h ) c (S t ) − ) − ) g n / h n (S t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) det(Id − Ad( c (S t ) ˇ p (ˇ h ) c (S t ) − ) − ) l n (S ) / h n (S t ) (cid:12)(cid:12)(cid:12) = d S ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) − Q α ∈ Ψ n ( l ) (cid:12)(cid:12)(cid:12) − ˇ h α (cid:12)(cid:12)(cid:12)Q α ∈ Ψ n (cid:12)(cid:12)(cid:12) − ˇ h α (cid:12)(cid:12)(cid:12) = d S ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) − Y α ∈ Ψ n ( η n (S )) (cid:12)(cid:12)(cid:12) − ˇ h α (cid:12)(cid:12)(cid:12) = d S ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) − (cid:12)(cid:12)(cid:12) − h h − n + (cid:12)(cid:12)(cid:12) n Y k = (cid:12)(cid:12)(cid:12) − h h − k (cid:12)(cid:12)(cid:12) n Y k = (cid:12)(cid:12)(cid:12) − h k h − n + (cid:12)(cid:12)(cid:12) Moreover, (cid:12)(cid:12)(cid:12) ∆ G (ˇ h ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ G(U ) (ˇ h ) (cid:12)(cid:12)(cid:12) = Q α ∈ Ψ n (cid:12)(cid:12)(cid:12) − ˇ h α (cid:12)(cid:12)(cid:12) Q α ∈ Ψ ( g (U )) (cid:12)(cid:12)(cid:12) − ˇ h α (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − h h − n + (cid:12)(cid:12)(cid:12) n Y k = (cid:12)(cid:12)(cid:12) − h h − k (cid:12)(cid:12)(cid:12) n Y j = (cid:12)(cid:12)(cid:12) − h k h − n + (cid:12)(cid:12)(cid:12) In particular,D( c (S t ) ˇ p (ˇ h ) c (S t ) − ) (cid:12)(cid:12)(cid:12) ∆ G(U ) (ˇ h ) (cid:12)(cid:12)(cid:12) D ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) (cid:12)(cid:12)(cid:12) ∆ G (ˇ h ) (cid:12)(cid:12)(cid:12) = d S ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) − (cid:12)(cid:12)(cid:12) − e − X n + (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y k = (cid:16) − h h − k (cid:17) n Y k = (cid:16) − h k h − n + (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − . Similarly, (cid:12)(cid:12)(cid:12) ∆ Ψ ′ (ˇ t ˇ a ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − h h − n + (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − e − X n + (cid:12)(cid:12)(cid:12) and it follows from Remark A.5 that Θ Π ′ m ( c (S ) ˇ p (ˇ t ˇ a ) c (S ) − ) = ± sgn( X n + ) e − ikX e k sgn( X n + ) X n + e X n + − e − X n + if m ≤ − X n + ) e − ikX e − k sgn( X n + ) X n + e X n + − e − X n + if m ≥ , i.e. (cid:12)(cid:12)(cid:12) ∆ Ψ ′ (ˇ t ˇ a ) (cid:12)(cid:12)(cid:12) Θ Π ′ m ( c (S ) ˇ p (ˇ t ˇ a ) c (S ) − ) = ± sgn( X n + ) e ikX e ( k − X n + ) X n + (1 − e − X n + ) if m ≤ − X n + ) e ikX e − ( k + X n + ) X n + (1 − e − X n + ) if m ≥ , Finally, using that (cid:12)(cid:12)(cid:12)(cid:12) det(Ad( c (S ) ˇ p (ˇ t ′ ˇ a ′ ) c (S ) − ) − − | η ′ (S1) (cid:12)(cid:12)(cid:12)(cid:12) = | − e − X n + | , d S ′ ( c (S ) ˇ p (ˇ t ˇ a ) c (S ) − ) = (cid:12)(cid:12)(cid:12) e − X n + (cid:12)(cid:12)(cid:12) and d ′ S ( c (S t ) ˇ p (ˇ h ) c (S t ) − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n + Y j = (cid:16) e iX − X n + h − j (cid:17) n Y j = (cid:16) h j e − iX − X n + (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e − (2 n − X n + (cid:12)(cid:12)(cid:12) , we get the second members of Equation (18). e now look at the first member of Equation (18). One can easily check that for every σ ∈ S n + , w = w , E , + w n + , E n + , ∈ W h ′ and y ∈ h n such that y = ( y , . . . , y n + ) = ( iX , . . . , iX n + ), we get: h y σ ( w ) , σ ( w ) i = X σ (1) | w , | + X σ (2 n + | w n + , | if σ (1) , σ (2 n + ∈ { , . . . , n }− X σ (1) | w , | − X σ (2 n + | w n + , | if σ (1) , σ (2 n + ∈ { n + , . . . , n + } X σ (1) | w , | − X σ (2 n + | w n + , | if σ (1) ∈ { , . . . , n } and σ (2 n + ∈ { n + , . . . , n + }− X σ (1) | w , | + X σ (2 n + | w n + , | if σ (1) ∈ { n + , . . . , n + } and σ (2 n + ∈ { , . . . , n } (see the proof of Proposition A.3 for en easier computation) and then Γ σ, S t = h n if { σ (1) , σ (2 n + } ⊆ S t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (2 n + > (cid:9) if σ (1) ∈ S t and σ (2 n + ∈ A t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (2 n + < (cid:9) if σ (1) ∈ S t and σ (2 n + ∈ B t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (1) > (cid:9) if σ (2 n + ∈ S t and σ (1) ∈ A t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (1) < (cid:9) if σ (2 n + ∈ S t and σ (1) ∈ B t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (1) > , X σ (2 n + > (cid:9) if { σ (1) , σ (2 n + } ∩ S t = {∅} and σ (1) , σ (2 n + ∈ A t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (1) > , X σ (2 n + < (cid:9) if { σ (1) , σ (2 n + } ∩ S t = {∅} and σ (1) ∈ A t , σ (2 n + ∈ B t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (1) < , X σ (2 n + > (cid:9) if { σ (1) , σ (2 n + } ∩ S t = {∅} and σ (2 n + ∈ A t , σ (1) ∈ B t (cid:8) y = ( y , . . . , y n + ) ∈ h n , X σ (1) < , X σ (2 n + < (cid:9) if { σ (1) , σ (2 n + } ∩ S t = {∅} and σ (1) , σ (2 n + ∈ B t Let ˇ h ∈ ˇH n (S t ) and h = p (ˇ h ) = ( h , . . . , h n + ) as in Notations 6.9. Assume that m ≥
1. Using Corollary A.4, weget that Θ Π ′ k ( ˇ p (ˇ h ′ )) ∆ ( ˇ p (ˇ h ′ )) = h ′ h ′ h ′− k h ′ − h ′ h ′− h ′− ( h ′ − h ′ ) = − h ′− k , ( h ′ = diag( h ′ , h ′ )) . Then, X σ ∈ W (H C n ) ε ( σ ) ∆ Φ (Z n ) ( σ (ˇ h )) ∆ Φ ( n ) (ˇ h ) lim r ∈ E σ, S tr → Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ ( ˇ p (ˇ h ′ ))det(1 − p (ˇ h ′ ) p (ˇ h )) σ W h d ˇ h ′ = i π ) X σ ∈ W (H C n ) ε ( σ ) ∆ Φ (Z n ) ( σ (ˇ h )) ∆ Φ ( n ) (ˇ h ) lim r ∈ E σ, S tr → Z H ′ h ′− h ′ m − (1 − h ′ ( rh ) − σ (1) )(1 − h ′ ( rh ) − σ (2 n + ) dh ′ dh ′ = i π ) X σ ∈ S n + ε ( σ ) h σ (1) h σ (2 n + ∆ Φ (Z n ) ( σ (ˇ h )) ∆ Φ ( n ) (ˇ h ) lim r ∈ E σ, S tr → Z S h ′− dh ′ h ′ − rh σ (1) Z S h ′ m − dh ′ h ′ − rh σ (2 n + = i π ) (2 n − X ≤ i , j ≤ n + h i h j ∆ ( i , j , ˇ h ) lim r ∈ E σ, S tr → Z S h ′− dh ′ h ′ − rh i Z S h ′ m − dh ′ h ′ − rh j = − n − X i ∈ J ( h ) ∪ B t X j ∈ K ( h ) ∪ A t h i h j ∆ ( i , j , ˇ h ) h − i h m − j = − n − X i ∈ J ( h ) ∪ B t X j ∈ K ( h ) ∪ A t h ni h n + mj n + Q k = k , i , j h k n + Q k = k , i ( h i − h k ) n + Q l = l , i , j ( h j − h l ) imilarly, if m =
0, we get: X σ ∈ W (H C n ) ε ( σ ) ∆ Φ (Z n ) ( σ (ˇ h )) ∆ Φ ( n ) (ˇ h ) lim r ∈ E σ, S tr → Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ ( ˇ p (ˇ h ′ ))det(1 − p (ˇ h ′ ) p (ˇ h )) σ W h d ˇ h ′ = i π ) (2 n − X ≤ i , j ≤ n + h i h j ∆ ( i , j , ˇ h ) lim r ∈ E σ, S tr → Z S h ′− dh ′ h ′ − rh i Z S h ′− dh ′ h ′ − rh j = n − X i , j ∈ J ( h ) ∪ B ti , j h i h j ∆ ( i , j , ˇ h ) h − i h − j = n − X i , j ∈ J ( h ) ∪ B ti , j h ni h nj n + Q k = k , i , j h k n + Q k = k , i ( h i − h k ) n + Q l = l , i , j ( h j − h l )Finally, if m ≤ −
1, it follows from Corollary A.4 that: Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ ( ˇ p (ˇ h ′ )) = h ′ h ′ h ′− m h ′ − h ′ h ′− h ′− ( h ′ − h ′ ) = − h ′− m , (cid:16) h ′ = ( h ′ , h ′ ) (cid:17) . Then, X σ ∈ W (H C n ) ε ( σ ) ∆ Φ (Z n ) ( σ (ˇ h )) ∆ Φ ( n ) (ˇ h ) lim r ∈ E σ, S tr → Z ˇH ′ Θ Π ′ m ( ˇ p (ˇ h ′ )) ∆ ( ˇ p (ˇ h ′ ))det(1 − p (ˇ h ′ ) p (ˇ h )) σ W h d ˇ h ′ = i π ) X σ ∈ W (H C n ) ε ( σ ) ∆ Φ (Z( n )) ( σ (ˇ h )) ∆ Φ ( n ) (ˇ h ) lim r ∈ E σ, S tr → Z H ′ h ′ m − h ′ (1 − h ′ ( rh ) − σ (1) )(1 − h ′ ( rh ) − σ (2 n + ) dh ′ dh ′ = i π ) X σ ∈ S n + ε ( σ ) h σ (1) h σ (2 n + ∆ Φ (Z( n )) ( σ (ˇ h )) ∆ Φ ( n ) (ˇ h ) lim r ∈ E σ, S tr → Z S h ′ m − dh ′ h ′ − rh σ (1) Z S h ′− dh ′ h ′ − rh σ (2 n + = n − X i ∈ K ( h ) ∪ A t X j ∈ J ( h ) ∪ B t h i h j ∆ ( i , j , ˇ h ) h k − i h − j = n − X i ∈ K ( h ) ∪ A t X j ∈ J ( h ) ∪ B t h n + mi h nj n + Q k = k , i , j h k n + Q k = k , i ( h i − h k ) n + Q l = l , i , j ( h j − h l ) , and the theorem follows. (cid:3) We finish this section with a Lemma concerning the formulas we got in Theorem 6.10.
Lemma 6.11.
For every h ∈ H n , S t , n Q k = (cid:16) − h h − k (cid:17) n Q k = (cid:16) − h k h − n + (cid:17) ∈ R , and its sign is constant on every Weylchamber.Proof. This result was obtained in [21, Lemma 6.9] for t =
1. We prove this lemma for t = t =
2. We get: n Y k = (cid:16) − h h − k (cid:17) n Y k = (cid:16) − h k h − n + (cid:17) = (1 − h h − )(1 − h h − n )(1 − h h − n + )(1 − h n h − n + ) n − Y k = (cid:16) − h h − k (cid:17) (cid:16) − h k h − n + (cid:17) . irstly, n − Y j = (cid:16) − h h − j (cid:17) (cid:16) − h j h − n + (cid:17) = n − Y j = (cid:16) − e iX − X n + e − iX j (cid:17) (cid:16) − e iX j e − iX − X n + (cid:17) = n − Y j = (cid:12)(cid:12)(cid:12) − e iX − X n + e − iX j (cid:12)(cid:12)(cid:12) . Moreover,(1 − h h − n )(1 − h n h − n + ) = (1 − e iX − X n + e − iX + X n )(1 − e iX + X n e − iX − X n + ) = − X − X ) e X n − X n + + e X n − X n + ) and(1 − h h − n + )(1 − h h − n ) = (1 − e iX − X n e − iX + X n + )(1 − e iX + X n + e − iX − X n ) = − X − X ) e X n + − X n + + e X n + − X n ) . so the lemma follows. (cid:3) A ppendix A. T he dual pair (G = U(1) , G ′ = U( p , q ))In [21, Proposition 6.4], the first author gave explicit formulas for the value of the character Θ Π ′ on the compactCartan f H ′ = f H ′ ( ∅ ) of e G ′ . Moreover, for p = q =
1, he computed the character Θ Π ′ on e H ′ (S ), where H ′ (S ) is thenon-compact Cartan subgroup of e U(1 ,
1) as in Equation 10. In this section, we recover the results proved in [21]using the results of Section 5 and get formulas for Θ Π ′ on every Cartan subgroup of f G ′ By keeping the notations of Section 5, we get V = C with the hermitian form ( · , · ) given by( u , v ) = uv , ( u , v ∈ V) , V ′ = M n ′ , ( C ), where n ′ = p + q , with the skew-hermitian form ( · , · ) ′ given by:( u , v ) = v t i Id p , q u , and W = V ⊗ C V ′ the symplectic space defined by h w , w ′ i = Re(tr( w ′∗ w )) = Im( w ′ t Id p , q w ) . Similarly, H = G = U(1) = { h ∈ C , | h | = } , J = i , h = R J , and the group GL C (W) is given by:GL C (W) = (cid:8) g ∈ GL(W) , J ′ g = g J ′ (cid:9) = G ′ C , g GL C (W) = n ˜ g = ( g , ξ ) , g ∈ GL C (W) , det( g ) = ξ o . Using that V ′ = V ′ ⊕ . . . ⊕ V ′ n ′ , with V ′ j = C E j , , and the embedding h C ∋ λ → ( λ, , . . . , ∈ h ′ C , we get thatW h = (cid:8) w = ( w , , , , ..., , w , ∈ C (cid:9) , Z ′ = G ′ h = ( g ′ ∈ G ′ , g ′ = λ X ! , λ ∈ U(1) , X ∈ GL( n ′ − , C ) ) . In particular, Φ ′ (Z ′ ) = n ± ( e i − e j ) , ≤ i < j ≤ n ′ o , where e j is the form defined in Notation 5.1. For every ˇ h ′ ∈ ˇH ′ C ,with h ′ = ( h ′ , . . . , h ′ n ′ ), we get: ∆ Φ ′ ( ˇ h ′ ) = Y α> (ˇ h ′ α − ˇ h ′− α ) = Y ≤ i < j ≤ n ′ ( h ′ i h ′− j − h ′− i h ′ j ) = n ′ Y i = h ′− n ′− i Y ≤ i < j ≤ n ′ ( h ′ i − h ′ j ) , and for every σ ∈ S n ′ , ∆ Φ ′ (Z ′ ) ( σ ( ˇ h ′ )) = n ′ Y i = h ′− n ′− σ ( i ) Y ≤ i < j ≤ n ′ ( h ′ σ ( i ) − h ′ σ ( j ) ) = ε ( σ ) n ′ Y i = i , σ (1) h ′− n ′− i Y ≤ i < j ≤ n ′ i , j , σ (1) ( h ′ i − h ′ j ) . n particular, ∆ Φ ′ (Z ′ ) ( σ ( ˇ h ′ )) ∆ Φ ′ ( ˇ h ′ ) = ε ( σ ) h ′ n ′− σ (1) n ′ Q i = i , σ (1) h ′ in ′ Q i = i , σ (1) ( h ′ σ (1) − h ′ i ) = ε ( σ ) h ′ n ′− σ (1) n ′ Q i = h ′ in ′ Q i = i , σ (1) ( h ′ σ (1) − h ′ i ) ( ˇ h ′ ∈ ˇH ′ C ) . Notation A.1.
As in Section 6, because the set of genuine representations of e U(1) is isomorphic to Z , we willdenote by Π m , m ∈ Z , the representations of R ( e U(1) , ω ). Using [16], we get that Π m (˜ h ) = ± h m + q − p . We will denoteby Π ′ m the corresponding representation of e U( p , q ) and by Θ Π ′ m its character. Proposition A.2.
For every S ⊆ Ψ ′ st n (see Appendix B), the value of the character Θ Π ′ m on f H ′ (S) reg is given by thefollowing formula: ∆ Φ ′ (ˇ h ′ ) Θ Π ′ m ( c (S) ˇ p (ˇ h ′ ) c (S) − ) = X σ ∈ W (H ′ C ) ( − u ε ( σ ) | W (Z ′ C , H ′ C ) | ∆ Φ ′ (Z ′ ) ( σ − (ˇ h ′ ))det − k ( σ − (ˇ h ′ )) W h lim r ∈ E σ, S r → Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))det − k (ˇ h )det(1 − p (ˇ h ) rp (ˇ h ′ )) σ W h d ˇ hfor every ˇ h ′ ∈ ˇH ′ regS .Proof. Let Ψ ∈ C ∞ c ( f G ′ ) such that supp( Ψ ) ⊆ f G ′ · e H ′ (S), we get:(19) Θ Π ′ m ( Ψ ) = tr( P Π m ◦ ω ( Ψ )) = Z f G ′ Z e U(1) Θ Π m ( ˜ g ) Θ ( ˜ g ˜ g ′ ) d ˜ g ! Ψ ( ˜ g ′ ) d ˜ g ′ = Z f G ′ Z ˇU(1) Θ Π m ( ˇ p (ˇ h )) Θ ( ˇ p (ˇ h ) ˜ g ′ ) d ˇ h ! Ψ ( ˜ g ′ ) d ˜ g ′ , where P Π m : H → H ( Π m ) is the projection onto the Π m -isotypic component given by P Π m = ω ( Θ Π m ) (see [28,Section 1.4.6]), i.e. as a generalized function on f G ′ , Θ Π ′ m ( ˜ g ′ ) = Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))) Θ ( ˇ p (ˇ h ) ˜ g ′ ) d ˜ g , ( ˜ g ′ ∈ f G ′ ) . Using Remark 5.4, we get: Θ Π ′ m ( Ψ ) = Z f G ′ Θ Π ′ m ( ˜ g ′ ) Ψ ( ˜ g ′ ) d ˜ g ′ = m S Z ˇH ′ S ε Ψ ′ S , R (ˇ h ′ ) ∆ Φ ′ (ˇ h ′ ) H S ( Θ Π ′ m Ψ )(ˇ h ′ ) d ˇ h ′ = m S Z ˇH ′ S ε Ψ ′ S , R (ˇ h ′ ) ∆ Φ ′ (ˇ h ′ ) Θ Π ′ m ( c (S) ˇ p (ˇ h ′ ) c (S) − ) H S Ψ (ˇ h ′ ) d ˇ h ′ . (20)Using Theorem 5.5, Equation (19) can be written as: Θ Π ′ m ( Ψ ) = Z ˇU(1) Θ Π m ( ˇ p (ˇ h )) Z f G ′ Θ ( ˇ p (ˇ h )˜ g ′ ) Ψ (˜ g ′ ) d ˜ g ′ d ˇ h = Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))det − k (ˇ h ) det k (ˇ h ) Z f G ′ Θ ( ˇ p (ˇ h )˜ g ′ ) Ψ ( ˜ g ′ ) d ˜ g ′ ! d ˇ h = Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))det − k (ˇ h ) X σ ∈ W (H ′ C ) 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 (ˇ h ) rp (ˇ h ′ )) σ W h ε Ψ ′ S , R (ˇ h ′ ) H S ( Ψ )(ˇ h ′ ) d ˇ h ′ d ˇ h = Z ˇH ′ S ε Ψ ′ S , R (ˇ h ′ ) X σ ∈ W (H ′ C ) M S ( σ ) ∆ Φ ′ (Z ′ ) ( σ − (ˇ h ′ ))det − k ( σ − (ˇ h ′ )) W h lim r ∈ E σ, S r → Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))det − k (ˇ h )det(1 − p (ˇ h ) rp (ˇ h ′ )) σ W h d ˇ h H S ( Ψ )(ˇ h ′ ) d ˇ h ′ . (21) The result follows by comparing Equations 20 and 21. (cid:3)
Without loss of generality, we assume that p ≤ q and keep the notations of Appendix B (see Equation (23)). Inparticular, for every h ′ ∈ H ′ S t , 0 ≤ t ≤ p , h ′ is of the form h ′ = ( h ′ , . . . , h ′ n ′ ) = diag( e iX − X p + , . . . , e iX t − X p + t , e iX t + , . . . , e iX p , e iX + X p + , . . . , e iX t + X p + t , e iX p + t + , . . . , e iX p + q ) , X j ∈ R , here S t = n e − e p + , . . . , e t − e p + t o . Proposition A.3.
The value of the character Θ Π ′ m is given, for every ˇ h ′ ∈ ˇH ′ S t , ≤ t ≤ p, by the formula: Θ Π ′ m ( c (S t ) ˇ p (ˇ h ′ ) c (S t ) − ) = ± C − t P j = j ∈ J ( h ′ ) h ′− m + p − j n ′ Q i = h ′ i ! n ′ Q i = i , j ( h ′ j − h ′ i ) − t P j = j ∈ K ( h ′ ) h ′− m + p − p + j n ′ Q i = h ′ i ! n ′ Q i = i , p + j ( h ′ p + j − h ′ i ) − p P i = t + h ′− m + p − i n ′ Q j = h ′ j ! n ′ Q j = j , i ( h ′ i − h ′ j ) if m ≤ − − q − p t P j = j ∈ K ( h ′ ) h ′− m + p − j n ′ Q i = h ′ i ! n ′ Q i = i , j ( h ′ j − h ′ i ) + t P j = j ∈ K ( h ′ ) h ′− m + p − p + j n ′ Q i = h ′ i ! n ′ Q i = i , p + j ( h ′ p + j − h ′ i ) + p + q P j = p + t + h ′− m + p − j n ′ Q i = h ′ i ! n ′ Q i = j , i ( h ′ j − h ′ i ) otherwisewhere C = p + q − , h ′ = ( h ′ , . . . , h ′ n ′ ) and K ( h ′ ) , J ( h ′ ) are given by:J ( h ′ ) = { j ∈ { , . . . , t } , sgn( X p + j ) = } , K ( h ′ ) = { j ∈ { , . . . , t } , sgn( X p + j ) = − } , To make the equation shorter, we will denote by C the constant C = ( − u ( p + q − . Proof.
We start by determining the space E σ, ∅ for σ ∈ S n ′ . For every w = w , E , and y ′ = ( y ′ , . . . , y ′ n ′ ) ∈ h ′ with y ′ j = iX ′ j , X ′ j ∈ R , we get: h y ′ σ ( w ) , σ ( w ) i = h y ′ ( w , E σ (1) , ) , w , E σ (1) , i = h ( w , y ′ σ (1) E σ (1) , ) , w , E σ (1) , i = Im( w , E ,σ (1) Id p , q w , y ′ σ (1) E σ (1) , ) = | w , | Im( y ′ σ (1) E ,σ (1) E σ (1) , ) if 1 ≤ σ (1) ≤ p −| w , | Im( y ′ σ (1) E ,σ (1) E σ (1) , ) if p + ≤ σ (1) ≤ p + q = X ′ σ (1) | w , | if 1 ≤ σ (1) ≤ p − X ′ σ (1) | w , | if p + ≤ σ (1) ≤ p + q In particular, Γ σ, ∅ = n y ′ ∈ h ′ , X ′ σ (1) > o if 1 ≤ σ (1) ≤ p n y ′ ∈ h ′ , X ′ σ (1) < o if p + ≤ σ (1) ≤ n ′ and then E σ, ∅ = exp( i Γ σ, ∅ ) = n h ′ ∈ H ′ C , h ′ = ( e − X ′ , . . . , e − X ′ n ′ ) , X ′ σ (1) > o if 1 ≤ σ (1) ≤ p n h ′ ∈ H ′ C , h ′ = ( e − X ′ , . . . , e − X ′ n ′ ) , X ′ σ (1) < o if p + < σ (1) ≤ n ′ More generally, for every σ ∈ S n , we get: Γ σ, S t = h ′ if σ (1) ∈ S t n y ∈ h ′ , X ′ σ (1) > o if σ (1) ∈ { t + , . . . , p } n y ∈ h ′ , X ′ σ (1) < o if σ (1) ∈ { p + t + , . . . , n } In particular,E σ, S = exp n h ′ ∈ H ′ C , h = diag( e − X ′ , . . . , e − X ′ n ) , X ′ i ∈ R o if σ (1) ∈ S t n h ′ ∈ H ′ C , h ′ = diag( e − X ′ , . . . , e − X ′ n ) , X ′ i ∈ R , X ′ σ (1) > o if σ (1) ∈ { t + , . . . , p } n h ′ ∈ H ′ C , h = diag( e − X ′ , . . . , e − X ′ n ) , X ′ i ∈ R , X ′ σ (1) < o if σ (1) ∈ { p + t + , . . . , n } Because the space E σ, S t only depends on σ (1), we will denote this space by E i , S t for a σ ∈ S n ′ such that σ (1) = i .We first assume that n ′ is even, i.e. k =
0. Then, according to Proposition A.2 and that det(1 − p (ˇ h ) rp (ˇ h ′ )) σ W h = − h ( rh ′ ) − σ (1) , we get (up to a constant): Θ Π ′ ( c (S t ) p (ˇ h ′ ) c (S t ) − ) = C X σ ∈ S n ′ ε ( σ ) ∆ Φ ′ (Z ′ ) ( σ − (ˇ h ′ )) ∆ Φ ′ (ˇ h ′ ) lim r → r ∈ E σ, S Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))1 − h ( rh ′ ) − σ (1) d ˇ h C t X j = h ′ n ′− j n ′ Q i = h ′ i ! n ′ Q i = i , j ( h ′ j − h ′ i ) Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))1 − he − iX + X p + d ˇ h + C t X j = h ′ n ′− p + j n ′ Q i = h ′ i ! n ′ Q i = i , p + j ( h ′ p + j − h ′ i ) Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))1 − he − iX j − X p + j d ˇ h + C p X j = t + h ′ n ′− j n ′ Q i = h ′ i ! n ′ Q i = j , i ( h ′ j − h ′ i ) lim r → < r < Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))1 − h ( rh ′ j ) − d ˇ h + C n ′ X j = p + t + h ′ n ′− j n ′ Q i = h ′ i ! n ′ Q i = j , i ( h ′ j − h ′ i ) lim r → r > Z ˇU(1) Θ Π m ( ˇ p (ˇ h ))1 − h ( rh ′ i ) − d ˇ h = − C t X j = h ′ n ′− j n ′ Q i = h ′ i ! n ′ Q i = i , j ( h ′ j − h ′ i ) 1 e − iX j + X p + j Z ˇU(1) Θ Π m ( ˇ p (ˇ h )) h − e iX j − X p + j d ˇ h − C t X j = h ′ n ′− p + j n ′ Q i = h ′ i ! n ′ Q i = i , p + j ( h ′ p + j − h ′ i ) 1 e − iX j − X p + j Z ˇU(1) Θ Π m ( ˇ p (ˇ h )) h − e iX j + X p + j d ˇ h − C p X j = t + h ′ n ′− j n ′ Q i = h ′ i ! n ′ Q i = j , i ( h ′ j − h ′ i ) lim r → < r < h ′ i Z ˇU(1) Θ Π m ( ˇ p (ˇ h )) h − rh ′ j d ˇ h − C n ′ X j = p + t + h ′ n ′− j n ′ Q i = h ′ i ! n ′ Q i = j , i ( h ′ j − h ′ i ) lim r → r > h ′ i Z ˇU(1) Θ Π m ( ˇ p (ˇ h )) h − rh ′ j d ˇ h = − C i π t X j = h ′ n ′− j n ′ Q i = h ′ in ′ Q i = i , j ( h ′ j − h ′ i ) 1 e − iX j + X p + j Z U(1) h − m − − q − p h − e iX j − X p + j dh − C i π t X j = h ′ n ′− p + j n ′ Q i = h ′ in ′ Q i = i , p + j ( h ′ p + j − h ′ i ) 1 e − iX j − X p + j Z U(1) h − m − − q − p h − e iX j + X p + j dh − C i π p X j = t + h ′ n ′− j n ′ Q i = h ′ in ′ Q i = j , i ( h ′ j − h ′ j ) lim r → < r < h ′ i Z U(1) h − m − − q − p h − rh ′ j dh − C i π n ′ X j = p + t + h ′ n ′− j n ′ Q i = h ′ in ′ Q i = j , i ( h ′ j − h ′ i ) lim r → r > h ′ i Z U(1) h − m − − q − p h − rh ′ j dh If − m − − q − p ≥
0, i.e. m ≤ − − q − p . Then, according to Lemma 6.8, we get: Θ Π ′ m ( c (S t ) p (ˇ h ′ ) c (S t ) − ) = − C i π t X j = j ∈ J ( h ′ ) h ′ n ′− j n ′ Q i = h ′ in ′ Q i = i , j ( h ′ j − h ′ i ) 1 e − iX j + X p + j Z U(1) h − m − − q − p h − e iX j − X p + j dh − C i π t X j = j ∈ K ( h ′ ) h ′ n ′− p + j n ′ Q i = h ′ in ′ Q i = i , p + j ( h ′ p + j − h ′ i ) 1 e − iX j − X p + j Z U(1) h − m − − q − p h − e iX j + X p + j dh − C i π p X j = t + h ′ n ′− j n ′ Q i = h ′ in ′ Q i = j , i ( h ′ j − h ′ i ) lim r → < r < h ′ i Z U(1) h − m − − q − p h − rh ′ i dh = − C t X j = j ∈ J ( h ′ ) h ′− m + p − j n ′ Q i = h ′ in ′ Q i = i , j ( h ′ j − h ′ i ) − C t X j = j ∈ K ( h ′ ) h ′− m + p − p + j n ′ Q i = h ′ in ′ Q i = i , p + j ( h ′ p + j − h ′ i ) − C p X j = t + h ′− m + p − j n ′ Q i = h ′ in ′ Q i = j , i ( h ′ j − h ′ i ) imilarly, if m > − − q − p , we get: Θ Π ′ ( c (S t ) p (ˇ h ′ ) c (S t ) − ) = − C i π t X j = j ∈ K ( h ′ ) h ′ n ′− j n ′ Q i = h ′ in ′ Q i = i , j ( h ′ j − h ′ i ) 1 e − iX j + X p + j Z U(1) h − m − − q − p h − e iX j − X p + j dh − C i π t X j = j ∈ J ( h ′ ) h ′ n ′− p + j n ′ Q i = h ′ in ′ Q i = i , p + j ( h ′ p + j − h ′ i ) 1 e − iX j − X p + j Z U(1) h − m − − q − p h − e iX j + X p + j dh − C i π n ′ X j = p + t + h ′ n ′− j n ′ Q i = h ′ in ′ Q i = j , i ( h ′ j − h ′ i ) lim r → r > h ′ i Z U(1) h − m − − q − p h − rh ′ j dh = C t X j = j ∈ J ( h ′ ) h ′− m + p − j n ′ Q i = h ′ in ′ Q i = i , j ( h ′ j − h ′ i ) + C t X j = j ∈ K ( h ′ ) h ′− m + p − p + j n ′ Q i = h ′ in ′ Q i = i , p + j ( h ′ p + j − h ′ i ) + C n ′ X j = p + t + h ′− m + p − j n ′ Q i = h ′ in ′ Q i = j , i ( h ′ j − h ′ i )The computations are similar if n ′ is odd. (cid:3) Corollary A.4.
The value of Θ Π ′ m on e H ′ = ˇ p ( ˇH ′∅ ) is given, up to a constant, by: Θ Π ′ m ( ˇ p (ˇ h ′ )) = ± C n ′ Q i = h ′ i p P i = h ′ i − m + p − Q j , i ( h ′ i − h ′ j ) if m ≤ − − q − p − n ′ Q i = h ′ i n ′ P i = p + h ′ i − m + p − Q j , i ( h ′ i − h ′ j ) otherwisewhere C ∈ R . This result was obtained in [21, Section 6].
Remark
A.5 . Assume that p = q =
1. Then, Θ Π ′ m (˜ h ′ ) = ± ( e i θ − X ) − m e X − e − X if m ≤ − X > − ( e i θ + X ) − m e X − e − X if m ≤ − X < e i θ + X ) − m e X − e − X if m ≥ X > − ( e i θ − X ) − m e X − e − X if m ≥ X < , where h ′ = e i θ ch( X ) sh( X )sh( X ) e i θ ch( X ) ! . We recover the results of [21, Section 7].A ppendix B. C artan subgroups for unitary groups
It is well-known that the number of non-conjugated Cartan subgroups of G = U( p , q ), up to equivalence, ismin( p , q ) + = U( p ) × U( q ) be the maximal compact subgroup of G and H be the (diagonal) compact Cartan subgroup ofK. We denote by h , k and g the Lie algebras of H, K and G respectively and h C , k C and g C their complexifications. e denote by ∆ = ∆ ( g C , h C ) be the set of roots, by ∆ c : = ∆ c ( k C , h C ) the set of compact roots and by ∆ n = ∆ \ ∆ c the set of non-compact roots. Similarly, we denote by Ψ a set of positive roots of ∆ and let Ψ c and Ψ n the subsetsof Ψ given by Ψ c = ∆ c ∩ Ψ and Ψ n = ∆ n ∩ Ψ . In particular, g C = M α ∈ ∆ g C ,α , where g C ,α = { X ∈ g C , [ H , X ] = α ( H ) X , H ∈ h C } . Notation B.1. (1) For every α ∈ ∆ , we fix X α ∈ g C ,α , Y α ∈ g C , − α and H α ∈ i h such that:[ H α , X α ] = X α , [ H α , Y α ] = − Y α , [ X α , Y α ] = H α , H α = − H α = H − α , and such that X α = − Y α if α ∈ ∆ c and X α = Y α if α ∈ ∆ n .(2) We say that α, β ∈ ∆ are strongly orthogonal if α , ± β and α ± β < ∆ . We denote by Ψ st n a maximalfamily of strongly orthogonal roots of Ψ n (i.e. a subset of Ψ n such that every pairs α, β ∈ Ψ st n are stronglyorthogonal).For every α ∈ Ψ st n , we denote by c ( α ) the element of GL( p + q , C ) given by: c ( α ) = exp (cid:18) π Y α − X α ) (cid:19) . For every subset S of Ψ st n , we denote by c (S) the element of GL( p + q , C ) defined by(22) 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 Cartan subgroup of Gand 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) . where H C = n diag( λ , . . . , λ p + q ) , λ i ∈ C o .Without loss of generality, we assume that p ≤ q . The set of roots ∆ is given by ∆ = n ± ( e i − e j ) , ≤ i < j ≤ p + q o ,where e i is the linear form on h C = C p + q given by e i ( λ , . . . , λ p + q ) = λ i . In this case, ∆ c = n ± ( e i − e j ) , ≤ i < j ≤ p o ∪ n ± ( e i − e j ) , p + ≤ i < j ≤ p + q o , ∆ n = n ± ( e i − e j ) , ≤ i ≤ p , p + ≤ j ≤ p + q o , and the set Ψ st n can be chosen as { e t − e p + t , ≤ t ≤ p } . In particular, H( ∅ ) = H and if S t = { e − e p + , . . . , e t − e p + t } , ≤ t ≤ p , we get:H(S t ) = exp p M j = t + i R E j , j ⊕ p + q M j = p + t + i R E j , j ⊕ t M j = i R (E j , j + E p + j , p + j ) ⊕ t M j = R (E j , p + j + E p + j , j ) , and(23) H S t = n h = diag( e iX − X p + , . . . , e iX t − X p + t , e iX t + , . . . , e iX p , e iX + X p + , . . . , e iX t + X p + t , e iX p + t + , . . . , e iX p + ) , X j ∈ R o . Remark
B.2 . As explained in [26, Proposition 2.16], two Cartan subalgebras h (S ) and h (S ), with S , S ⊆ Ψ st n ,are conjugate if and only if there exists an element of σ ∈ W sending S ∪ ( − S ) onto S ∪ ( − S ). ppendix C. T he character ε Let (W , h· , ·i ) be a real symplectic space, Sp(W) the corresponding group of isometries and f Sp(W) its metaplecticcover as in Equation (1).Let W = X ⊕ Y be a complete polarization of W. We denote by Z the subgroup of Sp(W) preserving both X andY. In particular, we get that Z = ( g
00 ( g − ) t ! , g ∈ GL(X) ) ≈ GL(X) . We define a double cover g GL(X) of GL(X) by g GL(X) = n ( g , η ) ∈ GL(X) × C × , η = det( g ) o , g GL(X) ∋ ( g , η ) → g ∈ GL(X) . As recalled in [2, Section 6], the restriction mapZ ∋ g → g | X ∈ GL(X)lifts to a group isomorphism e Z ∋ ˜ g → ( g | X , η ) ∈ g GL(X) , where η = η ( ˜ g ) is defined on Z c = n ˜ g ∈ e Z , det( g − W , o by the following formula: η ( ˜ g ) = Θ ( ˜ g ) | Θ ( ˜ g ) || det( g | X ) | . We denote by ε the function on e Z given by: ε : e Z ∋ ˜ g → ε ( ˜ g ) = η ( ˜ g ) | η ( ˜ g ) | ∈ C . One can easily prove that ε is a character of e Z with values in the set {± , ± i } such that(24) ε ( ˜ g ) = Θ ( ˜ g ) | Θ ( ˜ g ) | , (cid:16) ˜ g ∈ e Z c (cid:17) . Lemma C.1.
For every ˜ g ∈ e Z c , we get: Θ ( ˜ g ) = det( g | X ) − det
12 ( c ( g | X ) + ! . Proof.
Then, for every ˜ g ∈ e Z c , we get: Θ ( ˜ g ) = det( i ( g − − = ( − dim R ( W )2 det( g | X − − det( g | Y − − = det( g | X − − det(1 − g | Y ) − = det( g | X − − det(1 − g − | X ) − = det( g | X − − det(1 − g − | X ) − = det( g | X )det( g | X − − We have:12 (cid:0) c ( g | X ) + (cid:1) = (cid:16) ( g | X + g | X − − + (cid:17) = (cid:16) ( g | X + g | X − − + ( g | X − g | X − − (cid:17) = g | X ( g | X − − . Then, Θ ( ˜ g ) = det( g | X )det( g | X − − = (cid:16) det( g | X )det( g | X − − (cid:17) det( g | X ) − = det( g | X ) − det
12 ( c ( g | X ) + ! . (cid:3) e define by det − X ( ˜ g ) the following quantity:det − X ( ˜ g ) = Θ ( ˜ g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det
12 ( c ( g | X ) + !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − . In particular, for every ˜ g ∈ e Z c , we get:(25) ε ( ˜ g ) = Θ ( ˜ g ) | Θ ( ˜ g ) | = det − X ( ˜ g ) | det − X ( ˜ g ) | (cid:16) ˜ g ∈ e Z c (cid:17) . R eferences [1] Anne-Marie Aubert and Tomasz Przebinda. A reverse engineering approach to the Weil representation. Cent. Eur. J. Math. , 12(10):1500–1585, 2014.[2] Florent Bernon and Tomasz Przebinda. Normalization of the Cauchy Harish-Chandra integral.
J. Lie Theory , 21(3):615–702, 2011.[3] Florent Bernon and Tomasz Przebinda. The Cauchy Harish-Chandra integral and the invariant eigendistributions.
Int. Math. Res. Not.IMRN , (14):3818–3862, 2014.[4] Abderrazak Bouaziz. Intégrales orbitales sur les groupes de Lie réductifs.
Ann. Sci. École Norm. Sup. (4) , 27(5):573–609, 1994.[5] Thomas J. Enright. Analogues of Kostant’s u-cohomology formulas for unitary highest weight modules.
J. Reine Angew. Math. , 392:27–36, 1988.[6] Harish-Chandra. Representations of semisimple Lie groups. III. Characters.
Proc. Nat. Acad. Sci. U.S.A. , 37:366–369, 1951.[7] Harish-Chandra. Representations of a semisimple Lie group on a Banach space. I.
Trans. Amer. Math. Soc. , 75:185–243, 1953.[8] Harish-Chandra. Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions.
Acta Math. , 113:241–318,1965.[9] Harish-Chandra. Invariant eigendistributions on a semisimple Lie group.
Trans. Amer. Math. Soc. , 119:457–508, 1965.[10] Henryk Hecht. The characters of some representations of Harish-Chandra.
Math. Ann. , 219(3):213–226, 1976.[11] Sigurdur Helgason. Di ff erential geometry, Lie groups, and symmetric spaces , volume 34 of Graduate Studies in Mathematics . AmericanMathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original.[12] Takeshi Hirai. The Plancherel formula for SU( p , q ). J. Math. Soc. Japan , 22:134–179, 1970.[13] Lars Hörmander.
The analysis of linear partial di ff erential operators. I . Classics in Mathematics. Springer-Verlag, Berlin, 2003. Distribu-tion theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)].[14] Roger Howe. Preliminaries i. (unpublished).[15] Roger Howe. Transcending classical invariant theory. J. Amer. Math. Soc. , 2(3):535–552, 1989.[16] M. Kashiwara and M. Vergne. On the Segal-Shale-Weil representations and harmonic polynomials.
Invent. Math. , 44(1):1–47, 1978.[17] Anthony W. Knapp.
Lie groups beyond an introduction , volume 140 of
Progress in Mathematics . Birkhäuser Boston, Inc., Boston, MA,second edition, 2002.[18] Stephen S. Kudla. On the local theta-correspondence.
Invent. Math. , 83(2):229–255, 1986.[19] Hung Yean Loke and Jiajun Ma. Invariants and K -spectrums of local theta lifts. Compos. Math. , 151(1):179–206, 2015.[20] Allan Merino. Characters of some unitary highest weight representations via the theta correspondence.
J. Funct. Anal. , 279(8):108698,70, 2020.[21] Allan Merino. Transfer of characters in the theta correspondence with one compact member.
J. Lie Theory , 30(4):997–1026, 2020.[22] Allan Merino. Transfer of characters for discrete series representations of the unitary groups in the equal rank case via the cauchy-harish-chandra integral. https: // arxiv.org / abs / , 2021.[23] Tomasz Przebinda. Characters, dual pairs, and unipotent representations. J. Funct. Anal. , 98(1):59–96, 1991.[24] Tomasz Przebinda. A Cauchy Harish-Chandra integral, for a real reductive dual pair.
Invent. Math. , 141(2):299–363, 2000.[25] Tomasz Przebinda. The character and the wave front set correspondence in the stable range.
J. Funct. Anal. , 274(5):1284–1305, 2018.[26] Wilfried Schmid. On the characters of the discrete series. The Hermitian symmetric case.
Invent. Math. , 30(1):47–144, 1975.[27] David A. Vogan, Jr.
Representations of real reductive Lie groups , volume 15 of
Progress in Mathematics . Birkhäuser, Boston, Mass.,1981.[28] Nolan R. Wallach.
Real reductive groups. I , volume 132 of
Pure and Applied Mathematics . Academic Press, Inc., Boston, MA, 1988.D epartment of M athematics , N ational U niversity of S ingapore , B lock S17, 10, L ower K ent R idge R oad , S ingapore epublic of S ingapore Email address : [email protected]@nus.edu.sg