aa r X i v : . [ m a t h . DG ] N ov On the geometry of Siegel-Jacobi domains
S. Berceanu, A. GheorgheNational Institute for Nuclear Physics and Engineering,P.O.Box MG-6, RO-077125 Bucharest-Magurele, RomaniaE-mail: [email protected] 18, 2018
Abstract
We study the holomorphic unitary representations of the Jacobi groupbased on Siegel-Jacobi domains. Explicit polynomial orthonormal bases ofthe Fock spaces based on the Siegel-Jacobi disk are obtained. The scalarholomorphic discrete series of the Jacobi group for the Siegel-Jacobi diskis constructed and polynomial orthonormal bases of the representationspaces are given.
M.S.C. 2000:
Key words:
Jacobi group, Siegel-Jacobi domain, canonical automor-phy factor, canonical kernel function, Fock representation, scalar holo-morphic discrete series
The Jacobi groups are semidirect products of appropriate semisimple real alge-braic group of Hermitian type with Heisenberg groups [27], [13]. The semisimplegroups are associated to Hermitian symmetric domains that are mapped intoa Siegel upper half space by equivariant holomorphic maps [23]. The Jacobigroups are unimodular, nonreductive, algebraic groups of Harish-Chandra type.The Siegel-Jacobi domains are nonreductive symmetric domains associated tothe Jacobi groups by the generalized Harish-Chandra embedding [23], [13], [28]-[30].The holomorphic irreducible unitary representations of the Jacobi groupsbased on Siegel-Jacobi domains have been constructed by Berndt, B¨ocherer,Schmidt, and Takase [9], [8], [25]-[27] with relevant topics: Jacobi forms, auto-morphic forms, spherical functions, theta functions, Hecke operators, and Kugafiber varieties.Some coherent state systems based on Siegel-Jacobi domains have been in-vestigated in the framework of quantum mechanics, geometric quantization,dequantization, quantum optics, nuclear structure, and signal processing [12],[19], [24], [2]-[6]. 1his paper is organized as follows. In Section 2 we present explicit formulasfor the canonical automorphy factors and kernel functions of the Jacobi groupsand corresponding Siegel-Jacobi domains. In Section 3 we introduce a Fockspace of holomorphic functions on the Siegel-Jacobi disk. We obtain explicitpolynomial orthonormal bases for this space and the Fock spaces with innerproducts associated to points on the Siegel disk (Proposition 3.1 and Proposi-tion 3.2). In Section 4 we construct the scalar holomorphic discrete series ofthe Jacobi group for the Siegel-Jacobi disk (Proposition 4.1). We give polyno-mial orthonormal bases of the representation spaces (Proposition 4.2). Finally,we discuss the link between the coherent state systems based on Siegel-Jacobidomains and the explicit kernel functions of representation spaces for Jacobigroups.
Notation.
We denote by R , C , Z , and N the field of real numbers, the fieldof complex numbers, the ring of integers, and the set of non-negative integers,respectively. M mn ( F ) ≅ F mn denotes the set of all m × n matrices with entriesin the field F . M n ( F ) is identified with F n . Set M n ( F ) = M nn ( F ). For any A ∈ M mn ( F ), t A denotes the transpose matrix of A . For A ∈ M mn ( C ), ¯ A denotes the conjugate matrix of A and A † = t ¯ A . For A ∈ M n ( C ), the inequality A > A is positive definite. The identity matrix of degree n isdenoted by I n . Let O ( D ,W ) denote the space of all W -valued holomorphicfunctions on the connected complex manifold D equipped with the topology ofuniform convergence on compact sets. Here W is a finite dimensional Hilbertspace. Set O ( D ) = O ( D ,W ) for dim W = 1. In this paper we will use the words” unitary representation on a Hilbert space” to mean a continuous unitaryrepresentation on a complex separable Hilbert space. We begin with the definition of the Jacobi group given in [9], [27], [13]. Let H n be the Siegel upper half space of degree n consisting of all symmetric matricesΩ ∈ M n ( C ) with Im Ω >
0. Let Sp( n, R ) be the symplectic group of degree n consisting of all matrices σ ∈ M n ( R ) such that t σJ n σ = J n , where σ = (cid:18) a bc d (cid:19) , J n = (cid:18) I n − I n (cid:19) , (2.1)and a , b , c , d ∈ M n ( R ). The group Sp( n, R ) acts transitively on H n by σ Ω =( a Ω + b )( c Ω + d ) − , where σ ∈ Sp( n, R ) and Ω ∈ H n .Let G s be a Zariski connected semisimple real algebraic group of Hermitiantype. Let D = G s /K s be the associated Hermitian symmetric domain, where K s is a maximal compact subgroup of G . Suppose there exist a homomorphism ρ : G s → Sp( n, R ) and a holomorphic map τ : D → H n such that τ ( gz ) = ρ ( g ) τ ( z )for all g ∈ G s and z ∈ D . The Jacobi group G J is the semidirect product of G s and the Heisenberg group H [ V ] associated with the symplectic R -space V and2he nondegenerate alternating bilinear form D : V × V → A , where A is thecenter of H [ V ]. The multiplication operation of G J ≈ G s × V × A is defined by gg ′ = ( σσ ′ , ρ ( σ ) v ′ + v, κ + κ ′ + 12 D ( v, ρ ( σ ) v ′ ) , (2.2)where g = ( σ, v, κ ) ∈ G J , g ′ = ( σ ′ , v ′ , κ ′ ) ∈ G J .The Jacobi-Siegel domain associated to the Jacobi group G J is defined by D J = D × C N ∼ = G J / ( K s × A ), where dim V = 2 N .Let w be a fixed element of D and let I τ ( w ) be the complex structure on V corresponding to τ ( w ) ∈ H n . Let V C = V + ⊕ V − be the complexification of V , where V ± consists of all v ∈ V C such that I τ ( w ) v = ± i v . Then w ∈ D and v ∈ V C determine the element v w = v + − τ ( w ) v − of V + , where v = v + + v − , v ± ∈ V ± . G J is an algebraic group of Harish-Chandra type [27], [13], [23]. We recallthe definition of Harish-Chandra type groups [23].Let G be a Zariski connected R -group with Lie algebra g and let G C be thecomplexification of G . Suppose there are given a Zariski connected R -subgroup K of G with Lie algebra k and connected unipotent C -subgroups P ± of G C withLie algebras p ± . The group G is called of Harish-Chandra type if the followingconditions are satisfied:(HC 1) g C = p + + k C + p − is a direct sum of vector spaces, (cid:2) k C , p ± (cid:3) ⊂ p ± , and p + = p − ; (HC 2) the map P + × K C × P − → G C gives a holomorphic injectionof P + × K C × P − onto its open image P + K C P − ; (HC 3) G ⊂ P + K C P − and G ∩ K C P − = K .If g ∈ P + K C P − ⊂ G C , we denote by ( g ) + ∈ P + , ( g ) ∈ K C , ( g ) − ∈ P − thecomponents of g such that g = ( g ) + ( g ) ( g ) − .The identity connected component of a linear algebraic group H is denotedin the usual topology by ˚ H . The generalized Harish-Chandra embedding of thehomogeneous space D = ˚ G/ ˚ K into p + is defined by g ˚ K z , where g ∈ ˚ G , z ∈ p + and exp z = ( g ) + . Then the ˚ G -invariant complex structure of D isdetermined by the natural inclusion D ֒ → P + ⊂ G C / ( K C P − ). Let ( G C × p + ) ′ be the set of elements ( g, z ) ∈ G C × p + such that g exp z ∈ P + K C P − and let( p + × p + ) ′ be the set of elements ( z , z ) ∈ p + × p + such that (exp ¯ z ) − exp z ∈ P + K C P − .The canonical automorphy factor J : ( G C × p + ) ′ → K C and the canonicalkernel function K : ( p + × p + ) ′ → K C for G are defined by J ( g, z ) = ( g exp z ) , K ( z ′ , z ) = (((exp z ) − exp z ′ ) ) − , (2.3)where ( g, z ) ∈ ( G C × p + ) ′ and ( z ′ , z ) ∈ ( p + × p + ) ′ .According to [27], [13] (Corollary 4.5, Proposition 4.7, and equation (6.1)),we obtain Theorem 2.1 a) The Jacobi group G J acts transitively on D J by gx = ( σw, v σw + t ( cτ ( w ) + d ) − z ) , ρ ( σ ) = (cid:18) a bc d (cid:19) , (2.4)3 here g = ( σ, v, κ ) ∈ G J and x = ( w, z ) ∈ D J .b) The canonical automorphy factor J for the Jacobi group G J is given by J ( g, x ) = ( J ( σ, w ) , , J ( g, x )) , (2.5) where g = ( σ, v, κ ) ∈ G J , x = ( w, z ) ∈ D J , J is the canonical automorphyfactor for G s , and J ( g, x ) = κ + 12 D ( v, v σw ) + 12 D (2 v + ρ ( σ ) z, J ( σ, w ) z ) . (2.6) c) The canonical kernel function K for the Jacobi group G J is given by K ( x, x ′ ) = ( K ( w, w ′ ) , , K ( x, x ′ )) , (2.7) where x = ( w, z ) ∈ D J , x ′ = ( w ′ , z ′ ) ∈ D J , K is the canonical kernel functionfor G s , and K ( x, x ′ ) = D (2¯ z ′ + 12 τ ( w ′ ) z, qz )+ 12 D (¯ z ′ , qτ ( w )¯ z ′ ) , q = ρ ( K ( w, w ′ )) − . (2.8)The Heisenberg group H n ( R ) consists of all elements ( λ, µ, κ ), where λ, µ ∈ M n ( R ), κ ∈ R with the multiplication law( λ, µ, κ ) ◦ ( λ ′ , µ ′ , κ ′ ) = ( λ + λ ′ , µ + µ ′ , κ + κ ′ + λ t µ ′ − µ t λ ′ ) . (2.9)Let G Jn = Sp( n, R ) ⋉ H n ( R ) be the semidirect product of the symplectic groupSp( n, R ) and the Heisenberg group H n ( R ) endowed with the following multipli-cation law:( σ, ( λ, µ, κ )) · ( σ ′ , ( λ ′ , µ ′ , κ ′ )) = ( σσ ′ , ( λσ ′ , µσ ′ , κ ) ◦ ( λ ′ , µ ′ , κ ′ )) , (2.10)where ( λ, µ, κ ), ( λ ′ , µ ′ , κ ′ ) ∈ H n ( R ) and σ, σ ′ ∈ Sp( n, R ). The Jacobi group G Jn of degree n acts transitively on the Jacobi-Siegel space H Jn = H n × C n by g (Ω , ζ ) = (Ω g , ζ g ), where (Ω , ζ ) ∈ H Jn , g = ( σ, ( λ, µ, κ )) ∈ G Jn , σ is given by(2.1), and [29]Ω g = ( a Ω + b )( c Ω + d ) − , ζ g = ν ( c Ω + d ) − , ν = ζ + λ Ω + µ. (2.11)According with Theorem 2.1 and [22], we have Proposition 2.1
The canonical automorphy factor J and the canonical kernelfunction K for Sp( n, R ) are given by J ( σ, Ω) = (cid:18) t ( c Ω + d ) − c Ω + d (cid:19) , (2.12) K (Ω ′ , Ω) = (cid:18) − Ω ′ (Ω ′ − Ω) − (cid:19) , (2.13) where Ω , Ω ′ ∈ H n and σ ∈ Sp( n, R ) is given by (2.1) . he canonical automorphy factor θ = J ( g, (Ω , ζ )) for G Jn is given by θ = κ + λ t ζ + ν t λ − ν ( c Ω + d ) − c t ν, ν = ζ + λ Ω + µ, (2.14) where g = ( σ, ( λ, µ, κ )) ∈ G Jn , σ is given by (2.1) , and (Ω , ζ ) ∈ H Jn .The canonical automorphy kernel K for G Jn is given by K (( ζ ′ , Ω ′ ) , ( ζ, Ω)) = −
12 ( ζ ′ − ¯ ζ )(Ω ′ − ¯Ω ′ ) − ( t ζ ′ − t ¯ ζ ) , (2.15) where (Ω , ζ ) , (Ω ′ , ζ ′ ) ∈ H Jn . Let D n be the Siegel disk of degree n consisting of all symmetric matrices W ∈ M n ( C ) with I n − W ¯ W >
0. Let Sp( n, R ) ∗ be the multiplicative group ofall matrices ω ∈ M n ( C ) such that ω = (cid:18) p q ¯ q ¯ p (cid:19) , t p ¯ p − t ¯ qq = I n , t p ¯ q = t ¯ qp, p, q ∈ M n ( C ) . (2.16)Remark that Sp( n, R ) ∗ = Sp( n, C ) ∩ U( n, n ) for n >
1. Sp( n, R ) ∗ acts tran-sitively on D n by ωW = ( pW + q )(¯ qW + ¯ p ) − , where ω ∈ Sp( n, R ) ∗ and W ∈ D n . Let K n ∗ ∼ = U( n ) be the maximal compact subgroup of Sp( n, R ) ∗ consisting of all ω ∈ Sp( n, R ) ∗ given by (2.16 ) with p ∈ U( n ) and q = 0. Then D n ∼ =Sp( n, R ) ∗ / U( n ).Let G Jn ∗ be the Jacobi group consisting of all elements ( ω, ( α, κ )), where ω ∈ Sp( n, R ) ∗ , α ∈ C n , κ ∈ i R , and endowed with the multiplication law( ω ′ , ( α ′ , κ ′ ))( ω, ( α, κ )) = (cid:0) ω ′ ω, κ + κ ′ + β t ¯ α − ¯ β t α (cid:1) , (2.17)where ( ω, ( α, κ )) , ( ω ′ , ( α ′ , κ ′ )) ∈ G Jn ∗ β = α ′ p + ¯ α ′ ¯ q , and ω is given by (2.16).The Heisenberg group H n ( R ) ∗ consists of all elements ( I n , ( α, κ )) ∈ G Jn ∗ ,where ω ∈ Sp( n, R ) ∗ , α ∈ C n , κ ∈ i R . The center A ∗ ∼ = R of H n ( R ) ∗ consistsof all elements ( I n , (0 , κ )) ∈ G Jn ∗ with κ ∈ i R . According to [30], there existsan isomorphism Θ : G Jn → G Jn ∗ given by Θ( g ) = g ∗ , g = ( σ, ( λ, µ, κ )) ∈ G Jn , g ∗ = ( ω, ( α, κ )) ∈ G Jn ∗ , σ = (cid:18) a bc d (cid:19) , ω = (cid:18) p + p − p − p + (cid:19) , (2.18) p ± = 12 ( a ± d ) ± i2 ( b ∓ c ) , α = 12 ( λ + i µ ) , κ = − i κ . (2.19)Let D Jn = D n × C n ∼ = G Jn ∗ / (U( n ) × R ) be the Siegel-Jacobi disk of degree n . G Jn ∗ acts transitively on D Jn by g ∗ ( W, z ) = ( W g ∗ , z g ∗ ), where g ∗ = ( ω, ( α, κ )) ∈ G Jn ∗ , ( W, z ) ∈ D Jn , ω is given by (2.18), and [30] W g ∗ = ( pW + q )(¯ qW + ¯ p ) − , z g ∗ = ( z + αW + ¯ α )(¯ qW + ¯ p ) − . (2.20)5e now consider a partial Cayley transform of the Siegel-Jacobi disk D Jn onto the Siegel-Jacobi space H Jn which gives a partially bounded realization of H Jn [30]. The partial Cayley transform φ : D Jn → H Jn is defined byΩ = i( I n + W )( I n − W ) − , ζ = 2 i z ( I n − W ) − , (2.21)where ( ζ, Ω) = φ (( W, z )) and (
W, z ) ∈ D Jn . φ is a a biholomorphic map which satisfies gφ = φg ∗ for any g ∈ G Jn and g ∗ = Θ( g ) [30].The inverse partial Cayley transform φ − : H Jn → D Jn is given by W = (Ω − i I n )(Ω + i I n ) − , z = ζ (Ω + i I n ) − , (2.22)where ( W, z ) = φ − ((Ω , ζ )) ∈ D Jn and (Ω , ζ ) ∈ H Jn .According with Theorem 2.1, [22] and [30], we have Proposition 2.2
The canonical automorphy factor J ∗ and the canonical ker-nel function K ∗ for Sp( n, R ) ∗ are given by J ∗ ( ω, W ) = (cid:18) t (¯ qW + ¯ p ) −
00 ¯ qW + ¯ p (cid:19) , (2.23) K ∗ ( W ′ , W ) = (cid:18) I n − W ′ ¯ W t ( I n − W ′ ¯ W ) − (cid:19) , (2.24) where W, W ′ ∈ D n and ω ∈ Sp( n, R ) ∗ is given by (2.16). The canonical automorphy factor θ ∗ = J ( g ∗ , ( W, z )) for G J ∗ is given by θ ∗ = κ ∗ + z t α + ν ∗ t α − ν ∗ (¯ qW + ¯ p ) − ¯ q t ν ∗ , ν ∗ = z + αW + ¯ α, (2.25) where g ∗ = ( ω, ( α, κ )) ∈ G Jn ∗ , ω is given by (2.16) , and ( W, z ) ∈ D Jn .The canonical automorphy kernel for G J ∗ is given by K ∗ (( W ′ , z ′ ) , ( W, z )) = A ( W ′ , z ′ ; W, z ) , where ( W, z ) , ( W ′ , z ′ ) ∈ D Jn , and A ( W ′ , z ′ ; W, z ) = (¯ z + 12 z ′ ¯ W )( I n − W ′ ¯ W ) − t z ′ + 12 ¯ z ( I n − W ′ ¯ W ) − W ′ t ¯ z. (2.26) Let A ∗ ∼ = R be the center of the Heisenberg group H n ∗ ( R ). Given m ∈ R , let χ m be the central character of A ∗ defined by χ m ( κ ) = exp (2 π i mκ ), κ ∈ A ∗ .Suppose m > W ∈ D n we denote by F mW the Fock space of all functions Φ ∈O ( C n ) such that k Φ k mW < ∞ and the inner product is defined by [22](Φ , Ψ) mW = (2 πm ) n (cid:0) det(1 − W ¯ W ) (cid:1) − / (3.1) × Z C n Φ( z )Ψ( z ) exp( − πmA ( W, z )) dν ( z ) , C n is given by dν ( ζ ) = π − n n Y i =1 d Re ζ i d Im ζ i , (3.2)and A ( W, z ) = K ∗ (( W, z ) , ( W, z )) can be written as A ( W, z ) = (¯ z + 12 z ¯ W )( I n − W ¯ W ) − t z + 12 ¯ z ( I n − W ¯ W ) − W t ¯ z. (3.3)Remark that F m is the usual Bargmann space [1].We consider the Gaussian functions G U : D Jn → C , U ∈ C n , defined by G U ( W, Z ) = G ( U, Z, W ), where G ( U, Z, W ) = exp( U t Z + 12 U W t U ) = X s ∈ N n U s s ! P s ( Z, W ) (3.4)for all (
W, Z ) ∈ D Jn . We utilize the notation U s = n Y i =1 U s i i , s ! = n Y i =1 s i ! , | s | = n X i =1 s i , δ sr = n Y i =1 δ s i r i , (3.5)where U = ( U , ..., U n ) ∈ M n ( C ) ∼ = C n , s = ( s , ..., s n ) ∈ N n , and r =( r , ..., r n ) ∈ N n . The polynomials P s : D Jn → C , s ∈ N n , are exactly the match-ing functions studied by Neretin [17]. We express the homogeneous polynomial P s of degree | s | in the following compact form: P s ( Z, W ) = X a ∈ A n , ˜ a ≤ s s !2 ˆ a a !( s − ˜ a )! Z s − ˜ a W a , (3.6)where A n is the set of all symmetric matrices a = ( a ij ) ≤ i, j ≤ n with a ij ∈ N , W a = Y ≤ i ≤ j ≤ n W a ij ij , a ! = Y ≤ i ≤ j ≤ n a ij , ˜ a k = n X i =1 a ik , ˆ a = n X i =1 a ii , (3.7)and ˜ a ≤ s is equivalent with ˜ a i ≤ s i for 1 ≤ i ≤ n . Using the equations Z C n U s ¯ U r dν ( U ) = δ sr s ! , (3.8) Z C n G ( U, Z ′ , W ′ ) G ( ¯ U , ¯ Z, ¯ W ) dν ( U ) = det(1 − W ′ ¯ W ) − / exp A ( W ′ , z ′ ; W,z ) , (3.9)where A ( W ′ , z ′ ; W, z ) is defined by (2.26), we obtain (cid:0) det(1 − W ′ ¯ W ) (cid:1) − / exp A ( W ′ , z ′ ; W, z ) = X s ∈ N n s ! P s ( Z ′ , W ′ ) P s ( Z, W ) . (3.10)Equation (3.9) is given in [11] (Lemma 5).We now define the polynomials Φ Ws ∈ F mW , s ∈ N n , byΦ Ws ( z ) = 1 √ s ! P s (2 √ πmz, W ) , s ∈ N n . (3.11)7 roposition 3.1 Given W ∈ D n , the set of polynomials { Φ Ws | s ∈ N n } formsan orthonormal basis of the Fock space F mW . The kernel function of F mW admitsthe expansion (cid:0) det(1 − W ¯ W ) (cid:1) − / exp (2 πmA ( W, z ′ ; W, z )) = X s ∈ N n Φ Ws ( z ′ )Φ Ws ( z ) . (3.12) Proof . Given U ∈ C n and W ∈ D n , we define the function Ψ UW : C n → C such that Ψ UW ( z ) = G ( U, √ πmz, W ). Using the change of variables Z =2 √ πmz , we have k Ψ UW k mW = π − n det(1 − W ¯ W ) − / Z C n exp( B ( U, Z, W ) − A ( Z, W )) dν ( Z ) , (3.13)where B ( U, Z, W ) = U t Z + ¯ U Z † − U W t U −
12 ¯ U ¯ W U + . (3.14)Using the change of variables Z ′ = (1 − W ¯ W ) − / (cid:0) Z − ¯ U − W U (cid:1) , the relation dν ( Z ) = det(1 − W ¯ W ) dν ( Z ′ ), and the relation [1] Z C n exp( − Z ′ t Z ′ −
12 ( Z ′ ¯ W t Z ′ + Z ′ W Z ′† )) dν ( Z ′ ) = π n (cid:0) det(1 − W ¯ W ) (cid:1) − / , (3.15)we obtain k Ψ UW k mW = exp( U U † ). Then X s,r ∈ N n U s ¯ U r √ s ! r ! (Φ Ws , Φ Wr ) mW = X s ∈ N n s ! U s ¯ U s . (3.16)By comparing the coefficients of U s ¯ U r in the series of both sides of (3.16), wesee that (Φ Ws , Φ Wr ) mW = δ sr s ! , s, r ∈ N n . (3.17)Using (3.10) and (3.11), we obtain the expansion (3.12). (cid:4) We now introduce the set of polynomials f s : D Jn → C , s ∈ N n , defined by f s ( W, z ) = 1 √ s ! P s (2 √ πmz, W ) . (3.18)Let H ( D Jn ) be the complex linear subspace of all holomorphic functions f ∈ O ( D Jn ) with the basis { f s | s ∈ N n } . Let F m ( D Jn ) be the Hilbert space of allfunctions f ∈ O ( D Jn ) such that h f, f i m < ∞ , where the inner product h ., . i m isdefined such that the set { f s | s ∈ N n } is an orthonormal basis. We now prove Proposition 3.2 a) The generating function of the basis { f s | s ∈ N n } can beexpressed as exp(8 πmU t z + 12 U W t U ) = X s ∈ N n U s √ s ! f s ( W, z ) . (3.19)8 he kernel function of F m ( D Jn ) admits the expansion (cid:0) det(1 − W ′ ¯ W ) (cid:1) − / exp A ( W ′ , z ′ ; W, z ) = X s ∈ N n f s ( W ′ , z ′ ) f s ( W, z ) . (3.20) b) f ∈ O ( D Jn ) is a solution of the system of differential equations ∂ f∂z j ∂z k = 8 πm (1 + δ jk ) ∂f∂W jk , ≤ j ≤ k ≤ n, (3.21) if and only if f ∈ H ( D Jn ) .Proof. Using (3.4) and (3.18), we obtain (3.19). The generating function(3.19) satisfies (3.21). Then ∂ f s ∂z j ∂z k = 8 πm (1 + δ jk ) ∂f s ∂W jk , ≤ j ≤ k ≤ n, s ∈ N n . (3.22)Using (3.6) and (3.18), we obtain f s ( z, W ) = 1 √ s ! (2 √ πmz ) s + R s ( z, W ) , (3.23)where R s is a polynomial of degree | s | − z . Then there exists the change ofbasis { z s W a | s ∈ N , a ∈ A n } 7−→ { f s ( z, W ) W a | s ∈ N n , a ∈ A n } in O ( D Jn ). Let f ∈ O ( D Jn ). Then there exists the set { c s | c s ∈ O ( D n ) , s ∈ N n } such that f ( z, W ) = X s ∈ N n c s ( W ) f s ( W, z ) . (3.24)If f satisfies (3.21), then ∂c s /∂W jk = 0 for any 1 ≤ j ≤ k ≤ n, s ∈ N n . Then c s is constant for any s ∈ N n . Hence f ∈ O ( D Jn ). The inverse implication followsfrom (3.22). (cid:4) In the case n = 1, Proposition 3.2 has been obtained in [6]. Consider the Jacobi group G Jn . Let δ be a rational representation of GL ( n, C )such that δ | U( n ) is a scalar irreducible representation of the unitary group U( n )with highest weight k , k ∈ Z , and δ ( A ) = (det A ) k [31]. Let m ∈ R . Let χ = δ ⊗ ¯ χ m , where the central character χ m of A ∼ = R is defined by χ m ( κ ) =exp (2 π i mκ ), κ ∈ A . Any scalar holomorphic irreducible representation of G Jn ischaracterized by an index m and a weight k . Suppose m > k > n + 1 / H mk denote the Hilbert space of all holomorphic functions ϕ ∈ O ( H Jn )such that k ϕ k H Jn < ∞ with the inner product defined by [25]9 ϕ, ψ ) H Jn = C Z H Jn ϕ (Ω , ζ ) ψ (Ω , ζ ) K mk (Ω , ζ ) − dµ (Ω , ζ ) , (4.1)where C is a positive constant, (Ω , ζ ) ∈ H Jn and the G Jn -invariant measure on H Jn is given by dµ (Ω , ζ ) = (det Y ) − n − Y ≤ i ≤ n dξ i dη i Y ≤ j ≤ k ≤ n dX jk dY jk . (4.2)Here ξ = Re ζ, η = Im ζ, X = Re Ω , Y = Im Ω.The kernel function K mk is defined by [25] K mk (Ω , ζ ) = K mk ((Ω , ζ ) , (Ω , ζ )) = exp (cid:0) πmηY − t η (cid:1) (det Y ) k , (4.3) K mk (( ζ ′ , Ω ′ ) , ( ζ, Ω)) = (cid:18) det( i2 ¯Ω − i2 Ω ′ ) (cid:19) − k exp(2 π im K (( ζ ′ , Ω ′ ) , ( ζ, Ω))) , (4.4)where K is given by (2.15).Let π mk be the unitary representation of G Jn on H mk defined by [25] (cid:0) π mk ( g − ) ϕ (cid:1) (Ω , ζ ) = J mk ( g, (Ω , ζ )) ϕ (Ω g , ζ g ) , (4.5)where ϕ ∈ H mk , g ∈ G Jn , (Ω , ζ ) ∈ H Jn and (Ω g , ζ g ) ∈ H Jn is given by (2.11).The automorphic factor J mk for G Jn is defined by [25] J mk ( g, ( ζ, Ω)) = (det( c Ω + d )) − k exp(2 π i mθ ) , (4.6)where θ is given by (2.14) and σ is given by (2.1).Takase proved the following theorem [25], [26]: Theorem 4.1
Suppose k > n +1 / . Then H mk = { } and π mk is an irreducibleunitary representation of G Jn which is square integrable modulo center. Let H mk ∗ denote the complex pre-Hilbert space of all ψ ∈ O ( D Jn ) such that k ψ k D Jn < ∞ with the inner product defined by( ψ , ψ ) D Jn = C ∗ Z D Jn ψ ( W, z ) ψ ( W, z ) (cid:0) K mk ∗ ( W, z ) (cid:1) − d ν ( W, z ) , (4.7)where C ∗ is a positive constant, ( z, W ) ∈ D Jn , K mk ∗ ( W, z ) = (cid:0) det( I n − W ¯ W ) (cid:1) − k exp(8 πmA ( W, z )) , (4.8)where A is given by (3.3) and the G Jn -invariant measure on D Jn is [30] dν ( W, z ) = (det(1 − W ¯ W )) − n − n Y i =1 d Re z i d Im z i Y ≤ j ≤ k ≤ n d Re W jk d Im W jk. (4.9)According with [21], [30], and (2.26), the kernel function K mk ∗ is given by K mk ∗ ( W, z ) = K mk ∗ (( W, z ) , ( W, z )), where K mk ∗ (( z, W ) , ( z ′ , W ′ )) = (cid:0) det( I n − W ′ ¯ W ) (cid:1) − k exp (8 πmA ( W ′ , z ′ ; W, z )) . (4.10)10 emark 4.1 Using the coherent state method, Kramer, Saraceno, and Berceanuinvestigated the kernel (4.8) in the case 8 πm = 1 [12], [2]-[6].We now introduce the map g ∗ π mk ∗ ( g ∗ ), where π mk ∗ ( g ∗ ): H mk ∗ → H mk ∗ isdefined by (cid:0) π mk ∗ ( g − ∗ ) ψ (cid:1) ( z, W ) = J mk ∗ ( g ∗ , ( z, W )) ψ ( z g ∗ , W g ∗ ) , (4.11) ψ ∈ H mk ∗ , g ∗ = ( ω, ( α, κ )) ∈ G Jn ∗ , ( z, W ) ∈ D Jn , and ( z g ∗ , W g ∗ ) ∈ D Jn is givenby (2.20). The automorphic factor J mk ∗ for G Jn ∗ is defined by [21], [30] J mk ∗ ( g ∗ , ( z, W )) = exp(2 π i mθ ∗ ) (det(¯ qW + ¯ p )) − k (4.12)where θ ∗ is given by (2.25) and ω given by (2.16). Proposition 4.1
Suppose m > , k > n + 1 / , and C = 2 n ( n +3) C ∗ . Thena) H mk ∗ = { } and π mk ∗ is an irreducible unitary representation of G Jn ∗ onthe Hilbert space H mk ∗ which is square integrable modulo center.b) There exists the unitary isomorphism T mk ∗ : H mk ∗ → H mk given by ϕ (Ω , ζ ) = ψ ( W, z ) (det( I n − W )) k exp(4 πmz ( I n − W ) − t z ) , (4.13) where ψ ∈ H mk ∗ , ϕ = T mk ∗ ( ψ ) , ( W, z ) ∈ D Jn , (Ω , ζ ) = φ (( − W, z )) ∈ H Jn , and φ is given by (2.21) .The inverse isomorphism T mk : H mk → H mk ∗ is given by ψ ( W, z ) = ϕ (Ω , ζ ) (det( I n − iΩ)) k exp (cid:0) πmζ ( I n − iΩ) − t ζ (cid:1) , (4.14) where ψ ∈ H mk ∗ , ψ = T mk ( ϕ ) , (Ω , ζ ) ∈ H Jn , ( − W, z ) = φ − ((Ω , ζ )) ∈ D Jn , and φ − is given by (2.22) .c) The representations π mk and π mk ∗ are unitarily equivalent.Proof. Using the partial Cayley transform (2.21) and (2.22), we obtain Y = Im Ω = ( I n − W ) − ( I n − W ¯ W )( I n − ¯ W ) − , (4.15) η = Im ζ = z ( I n − W ) − + ¯ z ( I n − ¯ W ) − . (4.16)By (4.15) and (4.16), we obtain ηY − t η = 2 A ( z, − W ) − z ( I n − W ) − t z − ¯ z ( I n − ¯ W ) − t ¯ z, (4.17)where A is given by (3.3). Using (4.2), (4.9), (2.21), and (2.22), in the limitΩ → i I n and W →
0, we obtain dµ ( ζ, Ω) = 2 n ( n +3) dν ( z, W ) . (4.18)By (4.1), (4.7), (4.13), (4.14), the condition C = 2 n ( n +3) C ∗ , and the changeof variables W → − W , we get k ϕ k H Jn = k ψ k D Jn . From ζ ( I n − iΩ) − t ζ = − z ( I n − W ) − t z it is clear that (4.13) and (4.14) are equivalent. By Theorem4.1, a) and b) hold. Using (2.21), (2.22), (4.5), (4.11), (4.13), and (4.14), weobtain π mk T mk = T mk ∗ π mk ∗ . (cid:4) emark 4.2 Berndt, B¨ocherer and Schmidt constructed the holomorphic dis-crete series of the Jacobi group in the case n = 1 [8], [9].Let H k denote the complex Hilbert space of all holomorphic functions Φ ∈O ( D n ) such that k Φ k D n < ∞ , with the inner product defined by(Ψ , Ψ ) k = Z D n Ψ ( W )Ψ ( W ) (cid:0) det(1 − W ¯ W ) (cid:1) k − / dµ D n ( W ) , (4.19) dµ D n ( W ) = (cid:0) det(1 − W ¯ W ) (cid:1) − n − Y ≤ j ≤ k ≤ n d Re W jk d Im W jk . We have H k = { } for k > n +1 / { Q a | a ∈ A n } be an orthonormalpolynomial basis of H k .We introduce the polynomials F sa ( W, z ) = s (8 πm ) n C ∗ s ! P s ( √ πmz, W ) Q a ( W ) , s ∈ N n , a ∈ A n . (4.20) Proposition 4.2
The set of polynomials { F sa | s ∈ N n , a ∈ A n } forms an or-thonormal basis of H mk ∗ . The kernel function of H mk ∗ satisfies the expansion (det(1 − W ′ ¯ W )) − k exp A ( W ′ , z ′ , W, z ) = X s ∈ N n ,a ∈ A n F sa ( W ′ , z ′ ) F sa ( W, z ) . (4.21) Proof.
We introduce the functions F U : D Jn → C , U ∈ C n , such that F U ( W, z ) = G ( U, √ πmz, W ). Using (4.20) and the proof of Proposition 3.1,we have( F U ( W, z ) Q a ,F U ( W, z ) Q b ) D Jn = C ∗ (8 m ) n exp( U U † ) (4.22) × Z D n Q a ( W ) Q b ( W )det(1 − W ¯ W ) k − / dµ D n ( W ) , ( F sa , F rb ) D Jn = δ sr δ ab , s, r ∈ N n , a, b ∈ A n . (4.23)The Berezin kernel of H k is positive definite for k > n + 1 / (cid:0) det(1 − W ′ ¯ W ) (cid:1) − k +1 / = X a ∈ A n Q a ( W ′ ) Q a ( W ) . (4.24)Using (3.12) and (4.24), we obtain (4.21). (cid:4) Remark 4.3
In the case n = 1 and 8 πm = 1, the expansion (4.21) was obtainedin [3], using the coherent state method.12 emark 4.4 We now discuss the unitary representations of Jacobi groupsbased on Siegel-Jacobi domains in the language of coherent states [18]. Let Q ( H ) be the set of all one-dimensional projections of the Hilbert space H . Let P [ ψ ] denote the one-dimensional projection determined by ψ ∈ H\{ } . Theelements of Q ( H ) can be considered either as normal pure states of the vonNeumann algebra of bounded operators on H or as pure states of the C ∗ -algebraof compact operators on H [10]. The projective Hilbert space P ( H ) consists ofall one-dimensional complex linear subspaces of H . The space P ( H ) is a K¨ahlermanifold equipped with the usual Fubini-Study metric [10]. The space Q ( H )with relative w ∗ -topology is homeomorphic to P ( H ) with the manifold topology[10]. Then we can identify Q ( H ) with P ( H ).We recall an intrinsic definition of coherent state representations given in[15].Let G be a connected, simply connected Lie group and X a G -homogeneousspace which admits an invariant measure µ X . Let π be a continuous irreducibleunitary representation of G in the separable Hilbert space H . A family E = { E x | x ∈ X } of one-dimensional projections in H will be called a π - system ofcoherent states based on X if the following conditions are satisfied: 1) E g x = π ( g ) E x π ( g ) − for any g ∈ G and x ∈ X ; 2) there exists ψ ∈ H\{ } , suchthat R X |h ψ, π ( g ) ψ i| d µ X < ∞ . π is called a symplectic (K¨ahler) coherent staterepresentation if E and X are isomorphic symplectic (K¨ahler) manifolds and X is a symplectic (K¨ahler) submanifold of Q ( H ).Moscovici and Verona have been studied coherent state representations basedprecisely on the coadjoint orbit associated with π in the sense of geometricquantization [15]. The Schr¨odinger coherent state systems for the Heisenberggroup with one-dimensional center on the Fock spaces of holomorphic functionshave been obtained by Bargmann [1], Satake [20], [22], [23], and Lee [13].Lisiecki and Neeb investigated some K¨ahler coherent state representationsof Heisenberg groups and Jacobi groups with one-dimensional center [14],[16].The orbit method for the Heisenberg group and the Jacobi group with multi-dimensional center has been studied in detail by Yang [28].Let π be an irreducible unitary representation of the Jacobi group G J withthe Jacobi-Siegel domain D and the kernel function K : D × D → Hom(
W, W ).The representation space H consists of holomorphic functions taking their valuesin a finite dimensional Hilbert space W . For each x ∈ D and v ∈ W , weconsider the vectors K xv ∈ H given by K xv ( x ′ ) = K ( x, x ′ ) v for any x ′ ∈ D .Then { P [ K xv ] | x ∈ D , v ∈ W } is a π -system of coherent states. In particular,the π mk -system of coherent states based on H Jn and the π mk ∗ -system of coherentstates based on D Jn are determined by the explicit kernel functions given by(4.4) and (4.10), respectively. Acknowledgments.
The authors are indebted to the Organizing Committee of
The International Conference of Differential Geometry and Dynamical Systems , Uni-versity Politehnica of Bucharest, Romania, for the opportunity to report results atthe meeting. S. B. was partially supported by the CNCSIS-UEFISCSU project PNII-IDEI 454/2009, CNCSIS Cod ID-44, A.G. was partially supported by the CNCSIS- EFISCSU project PNII- IDEI 545/2009, CNCSIS Cod ID-1089 and both the authorshave been partially supported by the ANCS project program PN 09 37 01 02/2009.
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