Norm computation and analytic continuation of holomorphic discrete series

Transcription

1 Norm computation and analytic continuation of holomorphic discrete series Ryosuke Nakahama Graduate School of Mathematical Sciences, The University of Tokyo Introduction: Holomorphic discrete series of SU, Let : {w C : w < } Then G : SU, universal covering group acts on O by a b aw + b τ λ fw : cw + d λ f c d cw + d where λ C This action preserves the sesquilinear form f, h λ : λ fwhw w λ dw, π that is, τ λ gf, τ λgh λ f, h λ holds for any g G When Re λ >, this form converges for any polynomial f, h, and therefore τ λ defines a unitary representation of G when λ > This is called the holomorphic discrete series representation On the other hand, if Re λ, this integral diverges for any non-zero function f However, when Re λ > and f m a mw m, we can compute as f λ m m! λ m a m where λ m : λλ + λ + m, because r s λ π λ π w m w n w λ dw π λ δ mn s m s λ ds r m+n e iθm n r λ rdθdr δ mn λ Bm +, λ δ mn Γm + Γλ Γm + λ δ mn m! λ m This expression is available even when Re λ, and positive definite for λ > Therefore τ λ defines a unitary representation of G when λ > This example shows that if once the norm is explicitly computed, we can treat the analytic continuation of the holomorphic discrete series representation

2 Holomorphic discrete series of general Hermitian Lie group Let G be a Hermitian simple Lie group, that is, its maximal compact subgroup K has a non-discrete center We assume that G has a complexification G C We denote their Lie algebras by g, k, g C Then we can take z zk the center of k such that the eigenvalues of adz are +,, We denote the corresponding eigenspace decomposition by g C p + k C p We denote n dim C p +, r rank R G Then there exists a domain p + such that the following diagram commutes G/K G C /K C P p + exp Let τ, V be a holomorphic representation of K C, and let χ be the character of K C such that χexp t z e rt Then we have the following isomorphism Γ O G/K, G K V χ λ O, V where Γ O G/K, G K V χ λ is the space of holomorphic sections, and O, V is the space of V -valued holomorphic functions on G acts on O, V by the form τ λ gfw χµg, w λ τµg, w fg w for g G, w, where µ : G K C is a continuous map satisfying µgh, w µg, hwµh, w µk, w k g, h G, w, k K, w This action preserves the following norm: f λ,τ : c λ τbw π n fw, fw τ χbwλ p dw f O, V where p is an integer determined from g, and B : K C is a continuous map satisfying µg, wbwµg, w Bgw g G, w where is the inverse of the Cartan involution of KC This norm can be zero for any holomorphic function f, but if λ is sufficiently large, this does not occur Ir Example Let J, J I r Ir, and set I r G { g GLr, C : gj t g J, gj J ḡ } Spr, R, Then we have {w Symr, C : I ww is positive definite},

3 and G acts on O, V by A B τ λ fw : detcw + λ τ t Cw + f Aw + BCw + C This preserves f λ,τ : c λ π n τi ww fw, fw τ deti ww λ r+ dw We define another norm on Pp +, V f F,τ : π n fw τ e w dw f Pp +, V, p + where w is a suitable K-invariant norm on p + Let W Pp +, V be an irreducible subspace under the K-action ˆτkf w : τkfk w k K C, f Pp +, V, w p + Then since λ,τ and F,τ are both K-invariant, the ratio of two norms are constant on W The goal of this talk is to calculate this ratio under some conditions Remark Known results For several settings, this is already done by B Ørsted 98 for G SUr, r, scalar type J Faraut and A Korányi 99 for G any Hermitian Lie group, scalar type B Ørsted and G Zhang 994, 995 for G Spr, R, V C r, G SUr, r, V C C r, G SO 4r, V C r S Hwang, Y Liu and G Zhang 4 for G SUn,, V p C n C, q C n C 3 Main result We assume that G is of tube type Then there exists an e p + such that [e, ϑe] zk, [e, [ϑe, e]] e, where ϑ is the Cartan involution of g C Using this, we define c G : Inte πi 4 e ϑe G, L : c G K C, K L : L K We take a maximal abelian subalgebra h k We write the root system by g C, h C p + k C p h C in the usual fashion We define the positive system + g C, h C such that + p + Then there exist roots {γ,, γ r } p + such that, for a irreducible unitary representation V µ of K with lowest weight µ h C, V µ has a K L -invariant vector K L -spherical if and only if µ is of the form µ m γ + + m r γ r, m j Z, m m r 3

4 Let a l : h l Then a l is spanned by {γ al,, γ r al } For a representation V of K C, we denote by V the complex conjugate representation of V with respect to the real form L For s s,, s r C r and m m,, m r Z r, we write s m : s j d j j where d is the integer satisfying n r + d rr Now we state the main theorem Theorem 3 Let τ, V be an irreducible finite dimensional complex representation of K C with restricted lowest weight k γ k r al γ r We assume τ KL still remains irreducible Let W Pp +, V be a K-irreducible subspace If all the K L -spherical irreducible subspaces in W V have the same lowest weight n γ n r γ r, then for any f W, f λ,τ f F,τ holds under the suitable choice of c λ λ k λ n m j λ + k n k Example 3 Let G Spr, R Then we have K Ur, L GLr, R, K L Or, and a l h We identify h R r such that m,, m r R r is a lowest weight of a representation of K if and only if m m r, m j Z Then we have γ j e j We set V : V,,,,, k C r Then this remains irreducible when restrected }{{} k to K L, and V V holds We have Pp +, V V m V,,,,, }{{} m m r k m j Z m m r m j Z i < <i k V m+ei i k where e i i k is the vector whose i j -component is and otherwise, and V ν {} if ν is not dominant V m+ei i k V is decomposed as V m+ei i k V j < <j k V m+ei i k +e j j k, but since V ν is K L -spherical if and only if ν Z r, the only K L -spherical subspace in V m+ei i k V is V m+ei i k Therefore, for f V m+ei i k, we have Thus the norm f λ,τ ek f F,τ ek λ e k λ m+ei i k f λ,τ ek m m r m j Z i < <i k λ + e k m+ei i k e k λ ek λ m+ei i k f m+ei i k F,τ ek 4

5 is continued meromorphically, and for k r it becomes a unitary representation if and only if { k λ, k +,, r } r,, and when λ l l k,, r the corresponding representation space is H λ, V m m l i < <i k l m l+ m r m j Z Remark 33 Our result can be applied for G Spr, R, V k C r G SO 4r, V S k C r, S k C r V m+ei i i k G SUr, r, V W C, C W W : any irreducible representation of Ur G Spin, s, V C V k,,k,±k k Z G Spin, s +, V C V,, spin representation 4 Proof for G Spr, R case In this section we prove the theorem for G Spr, R case In this case we have K Ur, L GLr, R, K L Or, p + Symr, C, {w Symr, C : I ww is positive definite} Let τ, V be an irreducible unitary representation of K, and let W Pp +, V be a K-irreducible subspace under the action ˆτkf w : τkfk w t k k K C, f Pp +, V, w p + For f W we compute the ratio c λ f τi ww λ,τ π f n fw, fw τ deti ww λ r+ dw F,τ π n fw τ e trww dw p + under the condition V has the lowest weight k k,, k r V KL V Or still remains irreducible All the K L Or-spherical irreducible subspaces in W V W V have the same lowest weight n n,, n r 5

6 We set f λ,τ / f F,τ : R W,λ, and compute this ratio As a preparation, we set b t b b B : GLr, R : b jj >, j,, r, b r b r b rr : {x Symr, R : positive definite} Then we have a bijection t B, b b t b Also, for x and s s,, s r C r, we set s x : x s s x s s3 r x s r s r r x s r where k x : detx ij i,j k For example, for b b ij t B we have s b t b b s bs bs r rr Also we set Γ s : e trx s x detx r+ dy This integral converges if Re s j > j, and we have Γ s n+r e trbtb s b t b detb t b r+ b r b r b rr db n+r n+r t B t B j π n r and therefore we have Γ s + m Γ s e i j b ijb s bs bsr rr b b b rr r+ b r b r j j e b jjb s j j jj db jj Γ s j j, i j Γ s j + m j j Γ s j j e b ijdb ij j b rr db s j j s m m j Now we start the proof Let K W z, w Pp + p +, EndV be the reproducing kernel of W with respect to F,τ Then the ratio R W,λ is computed as c λ Tr V τi ww K W w, w deti ww λ r+ dw R W,λ Tr V K W w, we trww dw p + To proceed the computation, we prove the following lemma Lemma 4 For any integrable function f on p +, we have fwdw c fkx t kdkdx p + K where c is a constant 6

7 Proof By the integral formulas fwdw c p + K fxdx c K L fka t k i<j fk t t k i<j a i a j a j da da r dk, j t i t j dt dt r dk where a diaga,, a r, t diagt,, t r, we can deduce the lemma by putting a j t j By this lemma, this is equal to c λ Tr V τi kxk K W kx t k, kx t k deti x λ r+ dkdx I K R W,λ Tr V K W kx t k, kx t k e trx dkdx Since the reproducing kernel satisfies K K W kz t k, k w k τkk W z, wτk z, w p +, k K C, we have, for k K and x, Therefore we have Thus we get K W kx t k, kx t k τkk W x 4 xx 4, x 4 Ix 4 τk τkτx 4 KW x, eτx 4 τk Tr V K W kx, kx Tr V K W x, I, Tr V τi kxk K W kx t k, kx t k Tr V τi x K W x, I R W,λ c λ Now we set B W,λ : I I Tr V τi x K W x, I deti x λ r+ dkdx Tr V K W x, Ie trx dkdx Tr V τi x K W x, I deti x λ r+ dkdx, Γ W : Tr V K W x, Ie trx dkdx so that R W,λ c λ B W,λ /W W Now, we regard K W x, I Pp +, EndV as a function of x We define the action τ of K C on Pp +, EndV by τkf x : τkf k x t k τ t k k K C, F Pp +, EndV, x p + Then K W x, I is K L -invariant under τ Now we use the following isomorphism τ, Pp +, EndV ˆτ τ, Pp +, V V, 7

8 where τ, V is the complex conjugate of τ, V as a representation of L In this case L GLr, R, and we have τ, V τ, V Then under this isomorphism K W x, I sits in W V Since all the K L -spherical component in this tensor product has the lowest weight n, we conclude that K W x, I sits in V n, that is, there exists a function F Pp +, EndV such that, for any b b ij j i r t B, and τbf x b n b n b n r rr F x n b t b F x, τkf xdk K W x, I K L By using F, we rewrite B W,λ and Γ W B W,λ Tr V τi x F x deti x λ r+ dx, I Γ W Tr V F xe trx dx From now we calculate B W,λ For y, we set Iy : Tr V τy x F x dety x λ r+ dx y so that Ie B W,λ We take b t B such that y b t b, and set x bz t b Then Iy Tr V τbi z t b F bz t b detbi z t b λ r+ detb t b r+ dz I I I Tr V τi z τb F bz t bτ t b deti z λ r+ detb t r+ λ b dz Tr V τi z F z n b t b deti z λ r+ detb t r+ λ b dz r+ λ Ie n y dety B W,λ λ+n y dety r+ Now we calculate Iye try dy by two ways Iye try dy B W,λ e try λ+n y dety r+ dy BW,λ Γ λ + n, Iye try dy x,y x x,z Tr V e try Tr V τy x F x dety x λ r+ dxdy e trx+z Tr V τz F x detz λ r+ dxdz e trx F xdx e trz τz detz λ r+ dz Since e trz τz detz λ r+ dz commutes with K L -action and V KL is irreducible, this is the scalar operator on V Moreover, for a lowest weight vector v V and z b t b b t B, we have τz v, v τ τ t b b v, v τ τb v τ b k bkr rr v τ k b t b v τ k z v τ, 8

9 and e trz τz detz λ r+ dzv, v e trz k z detz λ r+ dz v τ Γ λ + k r + τ v τ Therefore e trz τz detz λ r+ dz Γ λ + k r+ idv, and B W,λ Γ Thus we have Now we determine c λ : References λ + k r+ Tr V e trx F xdx Γ Γ λ + n Γ λ + k r+ R W,λ c λ Γ λ + n Γ λ+k Then we have Γ λ+k r+ R W,λ Γ λ + k Γ λ + n Γ λ + k Γ λ Γ λ Γ λ + n λ k λ n λ + k r+ Γ W Γ λ + n [] J Faraut and A Korányi, Analysis on symmetric cones Oxford Mathematical Monographs Oxford Science Publications The Clarendon Press, Oxford University Press, New York, 994 [] S Hwang, Y Liu and G Zhang, Hilbert spaces of tensor-valued holomorphic functions on the unit ball of C n Pacific J Math 4 4, no, 33 3 [3] B Ørsted, Composition series for analytic continuations of holomorphic discrete series representations of SUn,n, Trans Amer Math Soc 6 98, no, [4] B Ørsted and G Zhang, Reproducing kernels and composition series for spaces of vector-valued holomorphic functions on tube domains J Funct Anal 4 994, no, 8 4 [5] B Ørsted and G Zhang, Reproducing kernels and composition series for spaces of vector-valued holomorphic functions Pacific J Math 7 995, no,