The Jacobi group and its holomorphic discrete series representations on Siegel-Jacobi domains

Abstract

This is the summary of a part of the talk delivered at the workshop held at the Tambov University in September 2012, reporting several results on Jacobi groups and its holomorphic representations published by the authors.

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Introduction The Jacobi group is the semidirect product of the real symplectic group with Heisenberg group of adequate dimension [9, 10]. Several generalizations are known [12, 20]. The Jacobi groups are unimodular, nonreductive, algebraic groups of Harish-Chandra type. The Siegel-Jacobi domains are nonreductive symmetric domains associated to the Jacobi groups by the generalized Harish-Chandra embedding [12, 16, 21, 22]. In [1] we have introduced Perelomov coherent states [13] defined on the Siegel-Jacobi disk. Similar constructions have been used previously [11, 14, 17]. The Jacobi group with applications in Quantum Mechanics has been investigated in a series of papers [2-7]. The present note is based manly on [5], where we have not used Perelomov coherent states. The problem of Berezin quantization [8], the fundamental conjecture for homogeneous K¨ahler manifolds, the classical and quantum evolution on Siegel-Jacobi domains, and the orthonormal base of polynomials in which the Bergman kernel is developed, all summarized in our the talk in accord with [2-7], are not included in this note. 1. Canonical automorphy factor and kernel function Let Hn be the Siegel upper half space of degree n consisting of all symmetric matrices 2 Mn(C) with Im > 0: The symplectic group Sp(n;R) of degree n consists of all matrices 2 M2n(R) such that t Jn = Jn; where = a b c d ; a; b; c; d 2 Mn(R); Jn = 0 In In 0 ; (1.1) The group Sp(n;R) acts transitively on Hn by = (a + b)(c + d) 1: Let Gs be a Zariski connected semisimple real algebraic group of Hermitian type. Let D =Gs=Ks be the associated Hermitian symmetric domain, where Ks is a maximal compact subgroup of Gs: Suppose there exist a homomorphism : Gs ! Sp(n;R) and a holomorphic map : D ! Hn such that (gz) = (g) (z) for all g 2 Gs and z 2 D: The Jacobi group GJ [9, 12, 20] is the semidirect product of Gs and the Heisenberg group H[V ] associated with the symplectic R-space V and the nondegenerate alternating bilinear form D : V -V ! A; where A is the center of H[V ]: The multiplication operation of GJ t Gs-V -A is defined by gg0 = ( 0; ( )v0 + v;{ + {0 + 1 2 D(v; ( )v0); where g = ( ; v; {) 2 GJ ; g0 = ( 0; v0;{0) 2 GJ : The Jacobi-Siegel domain associated to the Jacobi group GJ is defined by DJ = D - CN = GJ=(Ks - A); where dim V = 2N (cf. [9, 12, 20]). The definitions above are represented in the scheme below. GJ is an algebraic group of Harish-Chandra type [12, 16, 20]. Following [20] and [12], we obtain [5]. SIEGEL-JACOBI DOMAINS 347 GJ t Gs - V - A H[V ] Gs Sp(n;R) CN ,! DJ = D-CN - D = Gs=Ks Hn = Sp(n;R)=U(n) pr 1 pr 2 pr pr Theorem 1.1. a) The Jacobi group GJ acts transitively on DJ by gx = ( w; v w + t(c (w) + d) 1z); ( ) = a b c d ; where g = ( ; v; {) 2 GJ and x = (w; z) 2 DJ : b) The canonical automorphy factor J for the Jacobi group GJ is given by J(g; x) = (J1( ;w); 0; J2(g; x)); where J1 is the canonical automorphy factor for Gs; and J2(g; x) = { + 1 2 D(v; v w) + 1 2 D(2v + ( )z; J1( ;w)z): c) The canonical kernel function K for the Jacobi group GJ is given by K(x; x0) = (K1(w;w0); 0;K2(x; x0)); where K1 is the canonical kernel function for Gs; and K2(x; x0) = D(2z0 + 1 2 (w0)z; qz) + 1 2 D(z0; q (w)z0); q = (K1(w;w0)) 1: The Heisenberg group Hn(R) consists of all elements ( ; ; ); where ; 2 M1 n(R); 2 R; with the multiplication law ( ; ; ) ( 0; 0; 0) = ( + 0; + 0; + 0+ t 0 t 0): Let GJ n = Sp(n;R) n Hn(R) endowed with the following multiplication law: ( ; ( ; ; )) ( 0; ( 0; 0; 0)) = ( 0; ( 0; 0; ) ( 0; 0; 0)): The Jacobi group GJ n of degree n acts transitively on the Jacobi-Siegel space HJ n = Hn-Cn by g( ; ) = ( g; g); where g = (a + b)(c + d) 1; g = (c + d) 1; = + + : (1.2) 348 S. Berceanu, A. Gheorghe P r o p o s i t i o n 1.1. The canonical automorphy factor J1 and the canonical kernel function K1 for Sp(n;R) are given by J1( ; ) = t(c + d) 1 0 0 c + d ; K1( 0; ) = 0 0 ( 0 ) 1 0 ; where ; 0 2 Hn and 2 Sp(n;R) is given by (1.1). The canonical automorphy factor = J2(g; ( ; )) for GJ n is given by = + t + t (c + d) 1c t ; = + + ; (1.3) where g = ( ; ( ; ; )) 2 GJ n; is given by (1.1), and ( ; ) 2 HJ n: The canonical automorphy kernel K2 for GJ n is given by K2 (( 0; 0); ( ; )) = 1 2 ( 0 )( 0 0 ) 1(t 0 t ): (1.4) Let Dn be the Siegel disk of degree n consisting of all symmetric matrices W 2 Mn(C) with In WW > 0: Let Sp(n;R) be the multiplicative group of all matrices ! 2 M2n(C) such that ! = p q q p ; tpp tqq = In; tpq = tqp; p; q 2 Mn(C): (1.5) The group Sp(n;R) acts transitively on Dn by !W = (pW + q)(qW + p) 1: Let Kn = U(n) be the maximal compact subgroup of Sp(n;R) consisting of all ! 2 Sp(n;R) given by (1.5) with p 2 U(n) and q = 0: Then Dn = Sp(n;R) =U(n): Let GJ n be the Jacobi group consisting of all elements (!; ( ; {)); where ! 2 Sp(n;R) ; 2 Cn; { 2 iR; and endowed with the multiplication law (!0; ( 0;{0))(!; ( ; {)) = !0!; + ; { + {0 + t t ; where (!; ( ; {)); (!0; ( 0;{0)) 2 GJ n = 0p + 0q; and ! is given by (1.5). The Heisenberg group Hn(R) consists of all elements (In; ( ; {)) 2 GJ n ; with 2 Cn; { 2 iR: The center A = R of Hn(R) consists of all elements (In; (0;{)) 2 GJ n ; { 2 iR: There exists an isomorphism - : GJ n ! GJ n given by -(g) = g ; g = ( ; ( ; ; )) 2 GJ n; g = (!; ( ; {)) 2 GJ n ; = a b c d ; ! = p+ p p p+ ; p = 1 2 (a d) i 2 (b c); = 1 2 ( + i ); { = i 2 : Let DJ n = Dn - Cn = GJ n =(U(n) - R) be the Siegel-Jacobi disk of degree n. GJ n acts transitively on DJ n by g (W; z) = (Wg ; zg ); where Wg = (pW + q)(qW + p) 1; zg = (z + W + )(qW + p) 1: (1.6) SIEGEL-JACOBI DOMAINS 349 We now consider a partial Cayley transform of the Siegel-Jacobi disk DJ n onto the Siegel- Jacobi space HJ n which gives a partially bounded realization of HJ n [22]. The partial Cayley transform : DJ n ! HJ n is defined by = i(In +W)(In W) 1; = 2 i z(In W) 1; (1.7) where ( ; ) = ((W; z)) and (W; z) 2 DJ n: Thr map is a biholomorphic map which satisfies g = g for any g 2 GJ n and g = -(g) [22]. The inverse partial Cayley transform 1 : HJ n ! DJ n is given by W = ( iIn)( + iIn) 1; z = ( + iIn) 1: (1.8) The situation is summarized in the diagram below. GJ n GJ n Hn(R) Sp(n;R) Sp(n;R) Hn(R) Cn ,! HJ n = Hn - Cn - Hn = Sp(n;R)=U(n) Dn ,! DJ n = Dn - Cn - Cn - : GJ n ! GJ n pr iso iso P r o p o s i t i o n 1.2. The canonical automorphy factor J1 and the canonical kernel function K1 for Sp(n;R) are given by J1 (!;W) = t(qW + p) 1 0 0 qW + p ; K1 (W0;W) = In W0W 0 0 t(In W0W) 1 ; where W;W0 2 Dn and ! 2 Sp(n;R) is given by (1.5). The canonical automorphy factor = J2(gn ; (W; z)) for GJ n is given by = + z t + t (qW + p) 1q t ; = z + W + ; (1.9) where g = (!; ( ; {)) 2 GJ n ; ! is given by (1.5), and (W; z) 2 DJ n: The canonical automorphy kernel for GJ n is given by K2 ((W0; z0); (W; z)) = A(W0; z0;W; z); where (W; z); (W0; z0) 2 DJ n; and A(W0; z0;W; z) = (z + 1 2 z0W)(In W0W) 1 tz0 + 1 2 z(In W0W) 1W0 tz: (1.10) 350 S. Berceanu, A. Gheorghe 2. Scalar holomorphic discrete series Consider the Jacobi group GJ n: Let be a rational representation of GL(n;C) such that jU(n) is a scalar irreducible representation of the unitary group U(n) with highest weight k; k 2 Z; and (A) = (detA)k [23]. Let m 2 R: Let = m; where the central character m of A = R is defined by m( ) = exp (2 im ) ; 2 A: Any scalar holomorphic irreducible representation of GJ n is characterized by an index m and a weight k: Suppose m > 0 and k > n + 1=2: Let Hmk denote the Hilbert space of all holomorphic functions ' 2 O(HJ n) such that k'kHJ n < 1 with the inner product defined by [18] ('; )HJ n = C Z HJ n '( ; ) ( ; )Kmk( ; ) 1d ( ; ); where C is a positive constant, ( ; ) 2 HJ n and the GJ n -invariant measure on HJ n is given by d ( ; ) = (det Y ) n 2 Y 16i6n d i d-i Y 16j6k6n dXjk dYjk: Here = Re ; - = Im ; X = Re ; Y = Im : The kernel function Kmk is defined by [18] Kmk( ; ) = Kmk(( ; ); ( ; )) = exp 4 m-Y 1 t- (det Y )k; Kmk(( 0; 0); ( ; ))= det( i 2 i 2 0) kexp(2 imK (( 0; 0); ( ; ))) ; where K is given by (1.4). Let mk be the unitary representation of GJ n on Hmk defined by [18] mk(g 1)' ( ; ) = J mk(g; ( ; ))'( g; g); where ' 2 Hmk; g 2 GJ n; ( ; ) 2 HJ n and ( g; g) 2 HJ n is given by (1.2). The automorphic factor J mk for GJ n is defined by [18] J mk(g; ( ; )) = (det(c + d)) k exp(2 im ); where is given by (1.3) and is given by (1.1). Takase proved the following theorem [18, 19]: Theorem 2.1. Suppose k > n + 1=2: Then Hmk 6= f0g and mk is an irreducible unitary representation of GJ n which is square integrable modulo center. Let Hmk denote the complex pre-Hilbert space of all 2 O(DJ n) such that k kDJ n < 1 with the inner product defined by ( 1; 2)DJ n = C Z DJ n 1(W; z) 2(W; z) Kmk (W; z) 1 d (W; z); SIEGEL{JACOBI DOMAINS 351 where C is a positive constant, (z;W) 2 DJ n; Kmk (W; z) = det(In WW) k exp(8 mA(W; z)); and A(W; z) = K2 ((W; z); (W; z)) can be written as A(W; z) = (z + 1 2 zW)(In WW) 1 tz + 1 2 z(In WW) 1W tz: and the GJ n -invariant measure on DJ n is [22] d (W; z)=(det(1 WW)) n 2 Yn i=1 d Rezi d Imzi Y 1 j6k6n d ReWjkd ImWjk: According with [15, 22], and (1.10), the kernel function Kmk is given by Kmk (W; z) = Kmk ((W; z); (W; z)); where Kmk ((z;W); (z0;W0))= det(In W0W) k exp (8 mA(W0; z0;W; z)) : We now introduce the map g 7 ! mk (g ); where mk (g ) : Hmk ! Hmk is defined by mk (g 1 ) (z;W) = Jmk (g ; (z;W)) (zg ;Wg ); 2 Hmk ; g = (!; ( ; {)) 2 GJ n , (z;W) 2 DJ n; and (zg ;Wg ) 2 DJ n is given by (1.6). The automorphic factor Jmk for GJ n is defined by [15, 22] Jmk (g ; (z;W)) = exp(2 im ) (det(qW + p)) k ; where is given by (1.9) and ! given by (1.5). P r o p o s i t i o n 2.1. Suppose m > 0; k > n + 1=2; and C = 2n(n+3)C : Then a) Hmk 6= f0g and mk is an irreducible unitary representation of GJ n on the Hilbert space Hmk which is square integrable modulo center. b) There exists the unitary isomorphism Tmk : Hmk ! Hmk given by '( ; ) = (W; z) (det(In W))k exp(4 mz(In W) 1 tz); where 2 Hmk ; ' = Tmk ( ); (W; z) 2 DJ n; ( ; ) = (( W; z)) 2 HJ n; and is given by (1.7). The inverse isomorphism Tmk : Hmk ! Hmk is given by (W; z) = '( ; ) (det(In i ))k exp 2 m (In i ) 1 t ; where 2 Hmk , = Tmk(') , ( ; ) 2 HJ n , ( W; z) = 1 (( ; )) 2 DJ n; and 1 is given by (1.8). c) The representations mk and mk are unitarily equivalent. Acknowledgements. Stefan Berceanu express his thanks to Professor V. Molchanov for inviting him at the Workshop ¾Harmonic analysis on homogeneous spaces and quantization¿ October, 2012, Tambov, Russia, and for the partial financial support to attend the meeting.