aa r X i v : . [ m a t h . N T ] J u l INVARIANT DIFFERENTIAL OPERATORS ONSIEGEL-JACOBI SPACE
JAE-HYUN YANG
Abstract.
For two positive integers m and n , we let H n be the Siegel upper halfplane of degree n and let C ( m,n ) be the set of all m × n complex matrices. In thisarticle, we study differential operators on the Siegel-Jacobi space H n × C ( m,n ) thatare invariant under the natural action of the Jacobi group Sp ( n, R ) ⋉ H ( n,m ) R on H n × C ( m,n ) , where H ( n,m ) R denotes the Heisenberg group. We give some explicitinvariant differential operators. We present important problems which are natural.We give some partial solutions for these natural problems. Introduction
For a given fixed positive integer n , we let H n = { Ω ∈ C ( n,n ) | Ω = t Ω , Im Ω > } be the Siegel upper half plane of degree n and let Sp ( n, R ) = { M ∈ R (2 n, n ) | t M J n M = J n } be the symplectic group of degree n , where F ( k,l ) denotes the set of all k × l matriceswith entries in a commutative ring F for two positive integers k and l , t M denotesthe transpose matrix of a matrix M and J n = (cid:18) I n − I n (cid:19) .Sp ( n, R ) acts on H n transitively by(1.1) M · Ω = ( A Ω + B )( C Ω + D ) − , where M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R ) and Ω ∈ H n . For two positive integers m and n , we consider the Heisenberg group H ( n,m ) R = (cid:8) ( λ, µ ; κ ) | λ, µ ∈ R ( m,n ) , κ ∈ R ( m,m ) , κ + µ t λ symmetric (cid:9) endowed with the following multiplication law (cid:0) λ, µ ; κ (cid:1) ◦ (cid:0) λ ′ , µ ′ ; κ ′ (cid:1) = (cid:0) λ + λ ′ , µ + µ ′ ; κ + κ ′ + λ t µ ′ − µ t λ ′ (cid:1) with (cid:0) λ, µ ; κ (cid:1) , (cid:0) λ ′ , µ ′ ; κ ′ (cid:1) ∈ H ( n,m ) R . We define the semidirect product of Sp ( n, R ) and H ( n,m ) R G J = Sp ( n, R ) ⋉ H ( n,m ) R endowed with the following multiplication law (cid:0) M, ( λ, µ ; κ ) (cid:1) · (cid:0) M ′ , ( λ ′ , µ ′ ; κ ′ ) (cid:1) = (cid:0) M M ′ , (˜ λ + λ ′ , ˜ µ + µ ′ ; κ + κ ′ + ˜ λ t µ ′ − ˜ µ t λ ′ ) (cid:1) with M, M ′ ∈ Sp ( n, R ) , ( λ, µ ; κ ) , ( λ ′ , µ ′ ; κ ′ ) ∈ H ( n,m ) R and (˜ λ, ˜ µ ) = ( λ, µ ) M ′ . Then G J acts on H n × C ( m,n ) transitively by(1.2) (cid:0) M, ( λ, µ ; κ ) (cid:1) · (Ω , Z ) = (cid:16) M · Ω , ( Z + λ Ω + µ )( C Ω + D ) − (cid:17) , where M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R ) , ( λ, µ ; κ ) ∈ H ( n,m ) R and (Ω , Z ) ∈ H n × C ( m,n ) . Wenote that the Jacobi group G J is not a reductive Lie group and that the homogeneousspace H n × C ( m,n ) is not a symmetric space. We refer to [1, 6, 22, 23, 24, 25, 27, 28,29, 30, 31] about automorphic forms on G J and topics related to the content of thispaper. From now on, for brevity we write H n,m = H n × C ( m,n ) , called the Siegel-Jacobispace of degree n and index m .The aim of this paper is to study differential operators on H n,m which are invari-ant under the natural action (1.2) of G J . The study of these invariant differentialoperators on the Siegel-Jacobi space H n,m is interesting and important in the aspectsof invariant theory, arithmetic and geometry. This article is organized as follows. InSection 2, we review differential operators on H n invariant under the action (1.1) of Sp ( n, R ). We let D ( H n ) denote the algebra of all differential operators on H n that areinvariant under the action (1.1). According to the work of Harish-Chandra [7, 8], wesee that D ( H n ) is a commutative algebra which is isomorphic to the center of the uni-versal enveloping algebra of the complexification of the Lie algebra of Sp ( n, R ). Webriefly describe the work of Maass [14] about constructing explicit algebraically inde-pendent generators of D ( H n ) and Shimura’s construction [18] of canonically definedalgebraically independent generators of D ( H n ). In Section 3, we study differentialoperators on H n,m invariant under the action (1.2) of G J . For two positive integers m and n , we let T n,m = (cid:8) ( ω, z ) | ω = t ω ∈ C ( n,n ) , z ∈ C ( m,n ) (cid:9) be the complex vector space of dimension n ( n +1)2 + mn. From the adjoint action ofthe Jacobi group G J , we have the natural action of the unitary group U ( n ) on T n,m given by(1.3) u · ( ω, z ) = ( u ω t u, z t u ) , u ∈ U ( n ) , ( ω, z ) ∈ T n,m . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 3
The action (1.3) of U ( n ) induces canonically the representation τ of U ( n ) on thepolynomial algebra Pol( T n,m ) consisting of complex valued polynomial functions on T n,m . Let Pol( T n,m ) U ( n ) denote the subalgebra of Pol( T n,m ) consisting of all polyno-mials on T n,m invariant under the representation τ of U ( n ), and D ( H n,m ) denote thealgebra of all differential operators on H n,m invariant under the action (1 .
2) of G J . Wesee that there is a canonically defined linear bijection of Pol( T n,m ) U ( n ) onto D ( H n,m )which is not multiplicative. We will see that D ( H n,m ) is not commutative. The mainimportant problem is to find explicit generators of Pol( T n,m ) U ( n ) and explicit gener-ators of D ( H n,m ). We propose several natural problems. We want to mention thatat this moment it is quite complicated and difficult to find the explicit generators of D ( H n,m ) and to express invariant differential operators on H n,m explicitly. In Section4, we gives some examples of explicit G J -invariant differential operators on H n,m thatare obtained by complicated calculations. In Section 5, we deal with the special case n = m = 1 in detail. We give complete solutions of the problems that are proposed inSection 3. In Section 6, we deal with the case that n = 1 and m is arbitrary. We givesome partial solutions for the problems proposed in Section 3. In the final section,using these invariant differential operators on the Siegel-Jacobi space, we discuss anotion of Maass-Jacobi forms. Acknowledgements:
This work was in part done during my stay at the Max-Planck-Institut f¨ur Mathematik in Bonn. I am very grateful for the hospitality and financialsupport. I also thank the National Research Foundation of Korea for its financialsupport. Finally I would like to give my hearty thanks to Don Zagier, EberhardFreitag, Rainer Weissauer, Hiroyuki Ochiai and Minoru Itoh for their interests in thiswork and fruitful discussions.
Notations:
We denote by Q , R and C the field of rational numbers, the field ofreal numbers and the field of complex numbers respectively. We denote by Z and Z + the ring of integers and the set of all positive integers respectively. The symbol “:=”means that the expression on the right is the definition of that on the left. For twopositive integers k and l , F ( k,l ) denotes the set of all k × l matrices with entries in acommutative ring F . For a square matrix A ∈ F ( k,k ) of degree k , tr( A ) denotes thetrace of A . For any M ∈ F ( k,l ) , t M denotes the transpose matrix of M . I n denotesthe identity matrix of degree n . For A ∈ F ( k,l ) and B ∈ F ( k,k ) , we set B [ A ] = t ABA.
For a complex matrix A , A denotes the complex conjugate of A . For A ∈ C ( k,l ) and B ∈ C ( k,k ) , we use the abbreviation B { A } = t ABA.
For a positive integer n , I n denotes the identity matrix of degree n . JAE-HYUN YANG Invariant Differential Operators on the Siegel Space
For a coordinate Ω = ( ω ij ) ∈ H n , we write Ω = X + i Y with X = ( x ij ) , Y = ( y ij )real. We put d Ω = (cid:0) dω ij (cid:1) and d Ω = (cid:0) dω ij (cid:1) . We also put ∂∂ Ω = (cid:18) δ ij ∂∂ω ij (cid:19) and ∂∂ Ω = (cid:18) δ ij ∂∂ω ij (cid:19) . Then for a positive real number A ,(2.1) ds n ; A = A tr (cid:16) Y − d Ω Y − d Ω (cid:17) is a Sp ( n, R )-invariant K¨ahler metric on H n (cf. [19, 20]), where tr( M ) denotes thetrace of a square matrix M . H. Maass [13] proved that the Laplacian of ds n ; A is givenby(2.2) ∆ n ; A = 4 A tr (cid:18) Y t (cid:18) Y ∂∂ Ω (cid:19) ∂∂ Ω (cid:19) . And dv n (Ω) = (det Y ) − ( n +1) Y ≤ i ≤ j ≤ n dx ij Y ≤ i ≤ j ≤ n dy ij is a Sp ( n, R )-invariant volume element on H n (cf. [20, p. 130]).For brevity, we write G = Sp ( n, R ) . The isotropy subgroup K at iI n for the action(1.1) is a maximal compact subgroup given by K = (cid:26)(cid:18) A − BB A (cid:19) (cid:12)(cid:12)(cid:12) A t A + B t B = I n , A t B = B t A, A, B ∈ R ( n,n ) (cid:27) . Let k be the Lie algebra of K . Then the Lie algebra g of G has a Cartan decomposition g = k ⊕ p , where g = (cid:26)(cid:18) X X X − t X (cid:19) (cid:12)(cid:12)(cid:12) X , X , X ∈ R ( n,n ) , X = t X , X = t X (cid:27) , k = (cid:26)(cid:18) X − YY X (cid:19) ∈ R (2 n, n ) (cid:12)(cid:12)(cid:12) t X + X = 0 , Y = t Y (cid:27) , p = (cid:26)(cid:18) X YY − X (cid:19) (cid:12)(cid:12)(cid:12) X = t X, Y = t Y, X, Y ∈ R ( n,n ) (cid:27) . The subspace p of g may be regarded as the tangent space of H n at iI n . The adjointrepresentation of G on g induces the action of K on p given by(2.3) k · Z = kZ t k, k ∈ K, Z ∈ p . Let T n be the vector space of n × n symmetric complex matrices. We let Ψ : p −→ T n be the map defined by(2.4) Ψ (cid:18)(cid:18) X YY − X (cid:19)(cid:19) = X + i Y, (cid:18) X YY − X (cid:19) ∈ p . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 5
We let δ : K −→ U ( n ) be the isomorphism defined by(2.5) δ (cid:18)(cid:18) A − BB A (cid:19)(cid:19) = A + i B, (cid:18) A − BB A (cid:19) ∈ K, where U ( n ) denotes the unitary group of degree n . We identify p (resp. K ) with T n (resp. U ( n )) through the map Ψ (resp. δ ). We consider the action of U ( n ) on T n defined by(2.6) h · ω = hω t h, h ∈ U ( n ) , ω ∈ T n . Then the adjoint action (2.3) of K on p is compatible with the action (2.6) of U ( n )on T n through the map Ψ . Precisely for any k ∈ K and Z ∈ p , we get(2.7) Ψ( k Z t k ) = δ ( k ) Ψ( Z ) t δ ( k ) . The action (2.6) induces the action of U ( n ) on the polynomial algebra Pol( T n ) and thesymmetric algebra S ( T n ) respectively. We denote by Pol( T n ) U ( n ) (cid:16) resp. S ( T n ) U ( n ) (cid:17) the subalgebra of Pol( T n ) (cid:16) resp. S ( T n ) (cid:17) consisting of U ( n )-invariants. The followinginner product ( , ) on T n defined by( Z, W ) = tr (cid:0)
Z W (cid:1) , Z, W ∈ T n gives an isomorphism as vector spaces(2.8) T n ∼ = T ∗ n , Z f Z , Z ∈ T n , where T ∗ n denotes the dual space of T n and f Z is the linear functional on T n definedby f Z ( W ) = ( W, Z ) , W ∈ T n . It is known that there is a canonical linear bijection of S ( T n ) U ( n ) onto the algebra D ( H n ) of differential operators on H n invariant under the action (1.1) of G . Identifying T n with T ∗ n by the above isomorphism (2.8), we get a canonical linear bijection(2.9) Θ n : Pol( T n ) U ( n ) −→ D ( H n )of Pol( T n ) U ( n ) onto D ( H n ). The map Θ n is described explicitly as follows. Similarlythe action (2.3) induces the action of K on the polynomial algebra Pol( p ) and thesymmetric algebra S ( p ) respectively. Through the map Ψ, the subalgebra Pol( p ) K ofPol( p ) consisting of K -invariants is isomorphic to Pol( T n ) U ( n ) . We put N = n ( n + 1).Let { ξ α | ≤ α ≤ N } be a basis of a real vector space p . If P ∈ Pol( p ) K , then(2.10) (cid:16) Θ n ( P ) f (cid:17) ( gK ) = " P (cid:18) ∂∂t α (cid:19) f g exp N X α =1 t α ξ α ! K ! ( t α )=0 , where f ∈ C ∞ ( H n ). We refer to [9, 10] for more detail. In general, it is hard toexpress Φ( P ) explicitly for a polynomial P ∈ Pol( p ) K .According to the work of Harish-Chandra [7, 8], the algebra D ( H n ) is generatedby n algebraically independent generators and is isomorphic to the commutative ring JAE-HYUN YANG C [ x , · · · , x n ] with n indeterminates. We note that n is the real rank of G . Let g C bethe complexification of g . It is known that D ( H n ) is isomorphic to the center of theuniversal enveloping algebra of g C .Using a classical invariant theory (cf. [11, 21], we can show that Pol( T n ) U ( n ) isgenerated by the following algebraically independent polynomials(2.11) q j ( ω ) = tr (cid:16)(cid:0) ωω (cid:1) j (cid:17) , ω ∈ T n , j = 1 , , · · · , n. For each j with 1 ≤ j ≤ n, the image Θ n ( q j ) of q j is an invariant differentialoperator on H n of degree 2 j . The algebra D ( H n ) is generated by n algebraicallyindependent generators Θ n ( q ) , Θ n ( q ) , · · · , Θ n ( q n ) . In particular,(2.12) Θ n ( q ) = c tr (cid:18) Y t (cid:18) Y ∂∂ Ω (cid:19) ∂∂ Ω (cid:19) for some constant c . We observe that if we take ω = x + i y ∈ T n with real x, y , then q ( ω ) = q ( x, y ) =tr (cid:0) x + y (cid:1) and q ( ω ) = q ( x, y ) = tr (cid:16)(cid:0) x + y (cid:1) + 2 x (cid:0) xy − yx ) y (cid:17) . It is a natural question to express the images Θ n ( q j ) explicitly for j = 2 , , · · · , n. We hope that the images Θ n ( q j ) for j = 2 , , · · · , n are expressed in the form of the trace as Φ( q ).H. Maass [14] found algebraically independent generators H , H , · · · , H n of D ( H n ).We will describe H , H , · · · , H n explicitly. For M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R ) and Ω = X + iY ∈ H n with real X, Y , we setΩ ∗ = M · Ω = X ∗ + iY ∗ with X ∗ , Y ∗ real . We set K = (cid:0) Ω − Ω (cid:1) ∂∂ Ω = 2 i Y ∂∂ Ω , Λ = (cid:0) Ω − Ω (cid:1) ∂∂ Ω = 2 i Y ∂∂ Ω ,K ∗ = (cid:0) Ω ∗ − Ω ∗ (cid:1) ∂∂ Ω ∗ = 2 i Y ∗ ∂∂ Ω ∗ , Λ ∗ = (cid:0) Ω ∗ − Ω ∗ (cid:1) ∂∂ Ω ∗ = 2 i Y ∗ ∂∂ Ω ∗ . Then it is easily seen that(2.13) K ∗ = t ( C Ω + D ) − t (cid:8) ( C Ω + D ) t K (cid:9) , (2.14) Λ ∗ = t ( C Ω + D ) − t (cid:8) ( C Ω + D ) t Λ (cid:9) NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 7 and(2.15) t (cid:8) ( C Ω + D ) t Λ (cid:9) = Λ t ( C Ω + D ) − n + 12 (cid:0) Ω − Ω (cid:1) t C. Using Formulas (2.13), (2.14) and (2.15), we can show that(2.16) Λ ∗ K ∗ + n + 12 K ∗ = t ( C Ω + D ) − t (cid:26) ( C Ω + D ) t (cid:18) Λ K + n + 12 K (cid:19)(cid:27) . Therefore we get(2.17) tr (cid:18) Λ ∗ K ∗ + n + 12 K ∗ (cid:19) = tr (cid:18) Λ K + n + 12 K (cid:19) . We set(2.18) A (1) = Λ K + n + 12 K. We define A ( j ) ( j = 2 , , · · · , n ) recursively by A ( j ) = A (1) A ( j − − n + 12 Λ A ( j − + 12 Λ tr (cid:0) A ( j − (cid:1) (2.19) + 12 (cid:0) Ω − Ω (cid:1) t n(cid:0) Ω − Ω (cid:1) − t (cid:0) t Λ t A ( j − (cid:1)o . We set(2.20) H j = tr (cid:0) A ( j ) (cid:1) , j = 1 , , · · · , n. As mentioned before, Maass proved that H , H , · · · , H n are algebraically independentgenerators of D ( H n ).In fact, we see that(2.21) − H = ∆ n ;1 = 4 tr (cid:18) Y t (cid:18) Y ∂∂ Ω (cid:19) ∂∂ Ω (cid:19) . is the Laplacian for the invariant metric ds n ;1 on H n . Conjecture.
For j = 2 , , · · · , n, Θ n ( q j ) = c j H j for a suitable constant c j . Example 2.1.
We consider the case n = 1 . The algebra Pol( T ) U (1) is generated bythe polynomial q ( ω ) = ω ω, ω = x + iy ∈ C with x, y real . Using Formula (2.10), we getΘ ( q ) = 4 y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) . Therefore D ( H ) = C (cid:2) Θ ( q ) (cid:3) = C [ H ] . JAE-HYUN YANG
Example 2.2.
We consider the case n = 2 . The algebra Pol( T ) U (2) is generated bythe polynomial q ( ω ) = tr (cid:0) ω ω (cid:1) , q ( ω ) = tr (cid:16)(cid:0) ω ω (cid:1) (cid:17) , ω ∈ T . Using Formula (2.10), we may express Θ ( q ) and Θ ( q ) explicitly. Θ ( q ) isexpressed by Formula (2.12). The computation of Θ ( q ) might be quite tedious. Weleave the detail to the reader. In this case, Θ ( q ) was essentially computed in [4],Proposition 6. Therefore D ( H ) = C (cid:2) Θ ( q ) , Θ ( q ) (cid:3) = C [ H , H ] . In fact, the center of the universal enveloping algebra U ( g C ) was computed in [4].G. Shimura [18] found canonically defined algebraically independent generatorsof D ( H n ). We will describe his way of constructing those generators roughly. Let K C , g C , k C , p C , · · · denote the complexication of K, g , k , p , · · · respectively. Then wehave the Cartan decomposition g C = k C + p C , p C = p + C + p − C with the properties[ k C , p ± C ] ⊂ p ± C , [ p + C , p + C ] = [ p − C , p − C ] = { } , [ p + C , p − C ] = k C , where g C = (cid:26)(cid:18) X X X − t X (cid:19) (cid:12)(cid:12)(cid:12) X , X , X ∈ C ( n,n ) , X = t X , X = t X (cid:27) , k C = (cid:26)(cid:18) A − BB A (cid:19) ∈ C (2 n, n ) (cid:12)(cid:12)(cid:12) t A + A = 0 , B = t B (cid:27) , p C = (cid:26)(cid:18) X YY − X (cid:19) ∈ C (2 n, n ) (cid:12)(cid:12)(cid:12) X = t X, Y = t Y (cid:27) , p + C = (cid:26)(cid:18) Z iZiZ − Z (cid:19) ∈ C (2 n, n ) (cid:12)(cid:12)(cid:12) Z = t Z ∈ C ( n,n ) (cid:27) , p − C = (cid:26)(cid:18) Z − iZ − iZ − Z (cid:19) ∈ C (2 n, n ) (cid:12)(cid:12)(cid:12) Z = t Z ∈ C ( n,n ) (cid:27) . For a complex vector space W and a nonnegative integer r , we denote by Pol r ( W )the vector space of complex-valued homogeneous polynomial functions on W of degree r . We put Pol r ( W ) := r X s =0 Pol s ( W ) . Ml r ( W ) denotes the vector space of all C -multilinear maps of W × · · · × W ( r copies)into C . An element Q of Ml r ( W ) is called symmetric if Q ( x , · · · , x r ) = Q ( x π (1) , · · · , x π ( r ) ) NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 9 for each permutation π of { , , · · · , r } . Given P ∈ Pol r ( W ), there is a unique elementsymmetric element P ∗ of Ml r ( W ) such that(2.22) P ( x ) = P ∗ ( x, · · · , x ) for all x ∈ W. Moreover the map P P ∗ is a C -linear bijection of Pol r ( W ) onto the set of allsymmetric elements of Ml r ( W ). We let S r ( W ) denote the subspace consisting of allhomogeneous elements of degree r in the symmetric algebra S ( W ). We note thatPol r ( W ) and S r ( W ) are dual to each other with respect to the pairing(2.23) h α, x · · · x r i = α ∗ ( x , · · · , x r ) ( x i ∈ W, α ∈ Pol r ( W )) . Let p ∗ C be the dual space of p C , that is, p ∗ C = Pol ( p C ) . Let { X , · · · , X N } be a basisof p C and { Y , · · · , Y N } be the basis of p ∗ C dual to { X ν } , where N = n ( n + 1). Wenote that Pol r ( p C ) and Pol r ( p ∗ C ) are dual to each other with respect to the pairing(2.24) h α, β i = X α ∗ ( X i , · · · , X i r ) β ∗ ( Y i , · · · , Y i r ) , where α ∈ Pol r ( p C ) , β ∈ Pol r ( p ∗ C ) and ( i , · · · , i r ) runs over { , · · · , N } r . Let U ( g C )be the universal enveloping algebra of g C and U p ( g C ) its subspace spanned by theelements of the form V · · · V s with V i ∈ g C and s ≤ p. We recall that there is a C -linearbijection ψ of the symmetric algebra S ( g C ) of g C onto U ( g C ) which is characterizedby the property that ψ ( X r ) = X r for all X ∈ g C . For each α ∈ Pol r ( p ∗ C ) we define anelement ω ( α ) of U ( g C ) by(2.25) ω ( α ) := X α ∗ ( Y i , · · · , Y i r ) X i · · · X i r , where ( i , · · · , i r ) runs over { , · · · , N } r . If Y ∈ p C , then Y r as an element of Pol r ( p ∗ C )is defined by Y r ( u ) = Y ( u ) r for all u ∈ p ∗ C . Hence ( Y r ) ∗ ( u , · · · , u r ) = Y ( u ) · · · Y ( u r ) . According to (2.25), we see that if α ( P t i Y i ) = P ( t , · · · , t N ) for t i ∈ C with a polynomial P , then(2.26) ω ( α ) = ψ ( P ( X , · · · , X N )) . Thus ω is a C -linear injection of Pol( p ∗ C ) into U ( g C ) independent of the choice ofa basis. We observe that ω (cid:0) Pol r ( p ∗ C ) (cid:1) = ψ ( S r ( p C )) . It is a well-known fact that if α , · · · , α m ∈ Pol r ( p ∗ C ), then(2.27) ω ( α · · · α m ) − ω ( α m ) · · · ω ( α ) ∈ U r − ( g C ) . We have a canonical pairing(2.28) h , i : Pol r ( p + C ) × Pol r ( p − C ) −→ C defined by(2.29) h f, g i = X f ∗ ( e X i , · · · , e X i r ) g ∗ ( e Y i , · · · , e Y i r ) , where f ∗ (resp. g ∗ ) are the unique symmetric elements of Ml r ( p + C ) (resp. Ml r ( p − C )),and { e X , · · · , e X e N } and { e Y , · · · , e Y e N } are dual bases of p + C and p − C with respect to the Killing form B ( X, Y ) = 2( n + 1) tr( XY ), e N = n ( n +1)2 , and ( i , · · · , i r ) runs over (cid:8) , · · · , e N (cid:9) r . The adjoint representation of K C on p ± C induces the representation of K C onPol r ( p ± C ). Given a K C -irreducible subspace Z of Pol r ( p + C ) , we can find a unique K C -irreducible subspace W of Pol r ( p − C ) such that Pol r ( p − C ) is the direct sum of W and the annihilator of Z . Then Z and W are dual with respect to the pairing (2.28).Take bases { ζ , · · · , ζ κ } of Z and { ξ , · · · , ξ κ } of W that are dual to each other. Weset(2.30) f Z ( x, y ) = κ X ν =1 ζ ν ( x ) ξ ν ( y ) ( x ∈ p + C , y ∈ p − C ) . It is easily seen that f Z belongs to Pol r ( p C ) K and is independent of the choice ofdual bases { ζ ν } and { ξ ν } . Shimura [18] proved that there exists a canonically de-fined set { Z , · · · , Z n } with a K C -irreducible subspace Z r of Pol r ( p + C ) (1 ≤ r ≤ n )such that f Z , · · · , f Z n are algebraically independent generators of Pol( p C ) K . We canidentify p + C with T n . We recall that T n denotes the vector space of n × n symmet-ric complex matrices. We can take Z r as the subspace of Pol r ( T n ) spanned by thefunctions f a ; r ( Z ) = det r ( t aZa ) for all a ∈ GL ( n, C ) , where det r ( x ) denotes the de-terminant of the upper left r × r submatrix of x . For every f ∈ Pol( p C ) K , we letΩ( f ) denote the element of D ( H n ) represented by ω ( f ). Then D ( H n ) is the polyno-mial ring C [ ω ( f Z ) , · · · , ω ( f Z n )] generated by n algebraically independent elements ω ( f Z ) , · · · , ω ( f Z n ) . Invariant Differential Operators on Siegel-Jacobi Space
The stabilizer K J of G J at ( iI n ,
0) is given by K J = n(cid:0) k, (0 , κ ) (cid:1) (cid:12)(cid:12) k ∈ K, κ = t κ ∈ R ( m,m ) o . Therefore H n,m ∼ = G J /K J is a homogeneous space of non-reductive type . The Liealgebra g J of G J has a decomposition g J = k J + p J , where g J = n(cid:0) Z, ( P, Q, R ) (cid:1) (cid:12)(cid:12) Z ∈ g , P, Q ∈ R ( m,n ) , R = t R ∈ R ( m,m ) o , k J = n(cid:0) X, (0 , , R ) (cid:1) (cid:12)(cid:12) X ∈ k , R = t R ∈ R ( m,m ) o , p J = n(cid:0) Y, ( P, Q, (cid:1) (cid:12)(cid:12) Y ∈ p , P, Q ∈ R ( m,n ) o . Thus the tangent space of the homogeneous space H n,m at ( iI n ,
0) is identified with p J . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 11 If α = (cid:18)(cid:18) X Y Z − X (cid:19) , ( P , Q , R ) (cid:19) and β = (cid:18)(cid:18) X Y Z − X (cid:19) , ( P , Q , R ) (cid:19) areelements of g J , then the Lie bracket [ α, β ] of α and β is given by(3.1) [ α, β ] = (cid:18)(cid:18) X ∗ Y ∗ Z ∗ − X ∗ (cid:19) , ( P ∗ , Q ∗ , R ∗ ) (cid:19) , where X ∗ = X X − X X + Y Z − Y Z ,Y ∗ = X Y − X Y + Y t X − Y t X ,Z ∗ = Z X − Z X + t X Z − t X Z ,P ∗ = P X − P X + Q Z − Q Z ,Q ∗ = P Y − P Y + Q t X − Q t X ,R ∗ = P t Q − P t Q + Q t P − Q t P Lemma 3.1. [ k J , k J ] ⊂ k J , [ k J , p J ] ⊂ p J . Proof.
The proof follows immediately from Formula (3.1). (cid:3)
Lemma 3.2.
Let k J = (cid:18)(cid:18) A − BB A (cid:19) , (0 , , κ ) (cid:19) ∈ K J with (cid:18) A − BB A (cid:19) ∈ K, κ = t κ ∈ R ( m,m ) and α = (cid:18)(cid:18) X YY − X (cid:19) , ( P, Q, (cid:19) ∈ p J with X = t X, Y = t Y ∈ R ( n,n ) , P, Q ∈ R ( m,n ) . Then the adjoint action of K J on p J is given by (3.2) Ad ( k J ) α = (cid:18)(cid:18) X ∗ Y ∗ Y ∗ − X ∗ (cid:19) , ( P ∗ , Q ∗ , (cid:19) , where X ∗ = AX t A − (cid:0) BX t B + BY t A + AY t B (cid:1) , (3.3) Y ∗ = (cid:0) AX t B + AY t A + BX t A (cid:1) − BY t B, (3.4) P ∗ = P t A − Q t B, (3.5) Q ∗ = P t B + Q t A. (3.6) Proof.
We leave the proof to the reader. (cid:3)
We recall that T n denotes the vector space of all n × n symmetric complex matrices.For brevity, we put T n,m := T n × C ( m,n ) . We define the real linear map Φ : p J −→ T n,m by(3.7) Φ (cid:18)(cid:18) X YY − X (cid:19) , ( P, Q, (cid:19) = (cid:0) X + i Y, P + i Q (cid:1) , where (cid:18) X YY − X (cid:19) ∈ p and P, Q ∈ R ( m,n ) . Let S ( n, R ) denote the additive group consisting of all n × n real symmetric matrices.Now we define the isomorphism θ : K J −→ U ( n ) × S ( n, R ) by(3.8) θ ( h, (0 , , κ )) = ( δ ( h ) , κ ) , h ∈ K, κ ∈ S ( n, R ) , where δ : K −→ U ( n ) is the map defined by (2.5). Identifying R ( m,n ) × R ( m,n ) with C ( m,n ) , we can identify p J with T n × C ( m,n ) . Theorem 3.1.
The adjoint representation of K J on p J is compatible with the naturalaction of U ( n ) × S ( n, R ) on T n,m defined by (3.9) ( h, κ ) · ( ω, z ) := ( h ω t h, z t h ) , h ∈ U ( n ) , κ ∈ S ( n, R ) , ( ω, z ) ∈ T n,m through the maps Φ and θ . Precisely, if k J ∈ K J and α ∈ p J , then we have thefollowing equality (3.10) Φ (cid:0) Ad (cid:0) k J (cid:1) α (cid:17) = θ (cid:0) k J (cid:1) · Φ( α ) . Here we regard the complex vector space T n,m as a real vector space.Proof. Let k J = (cid:18)(cid:18) A − BB A (cid:19) , (0 , , κ ) (cid:19) ∈ K J with (cid:18) A − BB A (cid:19) ∈ K, κ = t κ ∈ R ( m,m ) and α = (cid:18)(cid:18) X YY − X (cid:19) , ( P, Q, (cid:19) ∈ p J with X = t X, Y = t Y ∈ R ( n,n ) , P, Q ∈ R ( m,n ) . Then we have θ (cid:0) k J (cid:1) · Φ( α ) = (cid:0) A + i B, κ (cid:1) · (cid:0) X + i Y, P + i Q (cid:1) = (cid:0) ( A + iB )( X + iY ) t ( A + iB ) , ( P + iQ ) t ( A + iB ) (cid:1) = (cid:0) X ∗ + i Y ∗ , P ∗ + i Q ∗ (cid:1) = Φ (cid:18)(cid:18) X ∗ Y ∗ Y ∗ − X ∗ (cid:19) , ( P ∗ , Q ∗ , (cid:19) = Φ (cid:0) Ad (cid:0) k J (cid:1) α (cid:17) ( by Lemma . , NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 13 where X ∗ , Y ∗ , Z ∗ and Q ∗ are given by the formulas (3.3), (3.4), (3.5) and (3.6) respec-tively. (cid:3) We now study the algebra D ( H n,m ) of all differential operators on H n,m invariantunder the natural action (1.2) of G J . The action (3.9) induces the action of U ( n ) onthe polynomial algebra Pol n,m := Pol ( T n,m ) . We denote by Pol U ( n ) n,m the subalgebra ofPol n,m consisting of all U ( n )-invariants. Similarly the action (3.2) of K induces theaction of K on the polynomial algebra Pol (cid:0) p J (cid:1) . We see that through the identificationof p J with T n,m , the algebra Pol (cid:0) p J (cid:1) is isomorphic to Pol n,m . The following U ( n )-invariant inner product ( , ) ∗ of the complex vector space T n,m defined by (cid:0) ( ω, z ) , ( ω ′ , z ′ ) (cid:1) ∗ = tr (cid:0) ωω ′ (cid:1) + tr (cid:0) z t z ′ (cid:1) , ( ω, z ) , ( ω ′ , z ′ ) ∈ T n,m gives a canonical isomorphism T n,m ∼ = T ∗ n,m , ( ω, z ) f ω,z , ( ω, z ) ∈ T n,m , where f ω,z is the linear functional on T n,m defined by f ω,z (cid:0) ( ω ′ , z ′ ) (cid:1) = (cid:0) ( ω ′ , z ′ ) , ( ω, z ) (cid:1) ∗ , ( ω ′ , z ′ ) ∈ T n,m . According to Helgason ([10], p. 287), one gets a canonical linear bijection of S ( T n,m ) U ( n ) onto D ( H n,m ). Identifying T n,m with T ∗ n,m by the above isomorphism, one gets a nat-ural linear bijection Θ n,m : Pol U ( n ) n,m −→ D ( H n,m )of Pol U ( n ) n,m onto D ( H n,m ) . The map Θ n,m is described explicitly as follows. We put N ⋆ = n ( n + 1) + 2 mn . Let (cid:8) η α | ≤ α ≤ N ⋆ (cid:9) be a basis of p J . If P ∈ Pol (cid:0) p J (cid:1) K =Pol U ( n ) n,m , then(3.11) (cid:16) Θ n,m ( P ) f (cid:17) ( gK J ) = " P (cid:18) ∂∂t α (cid:19) f g exp N ⋆ X α =1 t α η α ! K J ! ( t α )=0 , where f ∈ C ∞ ( H n,m ). In general, it is hard to express Θ n,m ( P ) explicitly for apolynomial P ∈ Pol (cid:0) p J (cid:1) K . We refer to [10], p. 287.We present the following basic U ( n )-invariant polynomials in Pol U ( n ) n,m . q j ( ω, z ) = tr (cid:0) ( ω ω ) j +1 (cid:1) , ≤ j ≤ n − , (3.12) α ( j ) kp ( ω, z ) = Re (cid:0) z ( ωω ) j t z (cid:1) kp , ≤ j ≤ n − , ≤ k ≤ p ≤ m, (3.13) β ( j ) lq ( ω, z ) = Im (cid:0) z ( ωω ) j t z (cid:1) lq , ≤ j ≤ n − , ≤ l < q ≤ m, (3.14) f ( j ) kp ( ω, z ) = Re ( z ( ωω ) j ω t z ) kp , ≤ j ≤ n − , ≤ k ≤ p ≤ m, (3.15) g ( j ) kp ( ω, z ) = Im ( z ( ωω ) j ω t z ) kp , ≤ j ≤ n − , ≤ k ≤ p ≤ m, (3.16)where ω ∈ T n and z ∈ C ( m,n ) . We present some interesting U ( n )-invariants. For an m × m matrix S , we definethe following invariant polynomials in Pol U ( n ) n,m : m (1) j ; S ( ω, z ) = Re (cid:16) tr (cid:0) ωω + t zSz (cid:1) j (cid:17) , ≤ j ≤ n, (3.17) m (2) j ; S ( ω, z ) = Im (cid:16) tr (cid:0) ωω + t zSz (cid:1) j (cid:17) , ≤ j ≤ n, (3.18) q (1) k ; S ( ω, z ) = Re (cid:16) tr (cid:0) ( t z S z ) k (cid:1)(cid:17) , ≤ k ≤ m, (3.19) q (2) k ; S ( ω, z ) = Im (cid:16) tr (cid:0) ( t z S z ) k (cid:1)(cid:17) , ≤ k ≤ m, (3.20) θ (1) i,k,j ; S ( ω, z ) = Re (cid:16) tr (cid:0) ( ωω ) i ( t z S z ) k ( ωω + t z S z ) j (cid:1)(cid:17) , (3.21) θ (2) i,k,j ; S ( ω, z ) = Im (cid:16) tr (cid:0) ( ωω ) i ( t z S z ) k ( ωω + t z S z ) j (cid:1)(cid:17) , (3.22)where 1 ≤ i, j ≤ n and 1 ≤ k ≤ m .We define the following U ( n )-invariant polynomials in Pol U ( n ) n,m . r (1) jk ( ω, z ) = Re (cid:16) det (cid:0) ( ωω ) j ( t zz ) k (cid:1)(cid:17) , ≤ j ≤ n, ≤ k ≤ m, (3.23) r (2) jk ( ω, z ) = Im (cid:16) det (cid:0) ( ωω ) j ( t zz ) k (cid:1)(cid:17) , ≤ j ≤ n, ≤ k ≤ m. (3.24)We propose the following natural problems. Problem 1.
Find a complete list of explicit generators of Pol U ( n ) n,m . Problem 2.
Find all the relations among a set of generators of Pol U ( n ) n,m . Problem 3.
Find an easy or effective way to express the images of the above invariantpolynomials or generators of Pol U ( n ) n,m under the Helgason map Θ n,m explicitly. Problem 4.
Decompose Pol n,m into U ( n )-irreducibles. Problem 5.
Find a complete list of explicit generators of the algebra D ( H n,m ). Orconstruct explicit G J -invariant differential operators on H n,m . Problem 6.
Find all the relations among a set of generators of D ( H n,m ). Problem 7.
Is Pol U ( n ) n,m finitely generated ? Is D ( H n,m ) finitely generated ?Quite recently Minoru Itoh [12] solved Problem 1 and Problem 7. Theorem 3.2.
Pol U ( n ) n,m is generated by q j ( ω, z ) , α ( j ) kp ( ω, z ) , β ( j ) lq ( ω, z ) , f ( j ) kp ( ω, z ) and g ( j ) kp ( ω, z ) , where ≤ j ≤ n − , ≤ k ≤ p ≤ m and ≤ l < q ≤ m . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 15 Examples of Explicit G J -Invariant Differential Operators In this section we give examples of explicit G J -invariant differential operators onthe Siegel-Jacobi space and the Siegel-Jacobi disk.For g = (cid:0) M, ( λ, µ ; κ ) (cid:1) ∈ G J with M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R ) and (Ω , Z ) ∈ H n,m , we set Ω ∗ = M · Ω = X ∗ + i Y ∗ , X ∗ , Y ∗ real ,Z ∗ = ( Z + λ Ω + µ )( C Ω + D ) − = U ∗ + i V ∗ , U ∗ , V ∗ real . For a coordinate (Ω , Z ) ∈ H n,m with Ω = ( ω µν ) and Z = ( z kl ), we put d Ω , d Ω , ∂∂ Ω , ∂∂ Ω as before and set Z = U + iV, U = ( u kl ) , V = ( v kl ) real ,dZ = ( dz kl ) , dZ = ( dz kl ) ,∂∂Z = ∂∂z . . . ∂∂z m ... . . . ... ∂∂z n . . . ∂∂z mn , ∂∂Z = ∂∂z . . . ∂∂z m ... . . . ... ∂∂z n . . . ∂∂z mn . Then we can show that d Ω ∗ = t ( C Ω + D ) − d Ω( C Ω + D ) − , (4.1) dZ ∗ = dZ ( C Ω + D ) − (4.2) + (cid:8) λ − ( Z + λ Ω + µ )( C Ω + D ) − C (cid:9) d Ω( C Ω + D ) − ,∂∂ Ω ∗ = ( C Ω + D ) t (cid:26) ( C Ω + D ) ∂∂ Ω (cid:27) (4.3) +( C Ω + D ) t (cid:26)(cid:0) C t Z + C t µ − D t λ (cid:1) t (cid:18) ∂∂Z (cid:19)(cid:27) and(4.4) ∂∂Z ∗ = ( C Ω + D ) ∂∂Z . From [14, p. 33] or [20, p. 128], we know that(4.5) Y ∗ = t ( C Ω + D ) − Y ( C Ω + D ) − = t ( C Ω + D ) − Y ( C Ω + D ) − . Using Formulas (4.1), (4.2) and (4.5), the author [29] proved that for any twopositive real numbers A and B , ds n,m ; A,B = A tr (cid:16) Y − d Ω Y − d Ω (cid:17) + B (cid:26) tr (cid:16) Y − t V V Y − d Ω Y − d Ω (cid:17) + tr (cid:16) Y − t ( dZ ) dZ (cid:17) − tr (cid:16) V Y − d Ω Y − t ( dZ ) (cid:17) − tr (cid:16) V Y − d Ω Y − t ( dZ ) (cid:17)(cid:27) is a Riemannian metric on H n,m which is invariant under the action (1.2) of G J . The following lemma is very useful for computing the invariant differential opera-tors. H. Maass [13] observed the following useful fact.
Lemma 4.1. (a) Let A be an m × n matrix and B an n × l matrix. Assume that theentries of A commute with the entries of B . Then t ( AB ) = t B t A. (b) Let A, B and C be a k × l , an n × m and an m × l matrix respectively. Assumethat the entries of A commute with the entries of B . Then t ( A t ( BC )) = B t ( A t C ) . Proof.
The proof follows immediately from the direct computation. (cid:3)
Using Formulas (4.3), (4.4), (4.5) and Lemma 4.1, the author [29] proved that thefollowing differential operators M and M on H n,m defined by(4.6) M = tr (cid:18) Y ∂∂Z t (cid:18) ∂∂Z (cid:19) (cid:19) and M = tr (cid:18) Y t (cid:18) Y ∂∂ Ω (cid:19) ∂∂ Ω (cid:19) + tr (cid:18) V Y − t V t (cid:18) Y ∂∂Z (cid:19) ∂∂Z (cid:19) (4.7) + tr (cid:18) V t (cid:18) Y ∂∂ Ω (cid:19) ∂∂Z (cid:19) + tr (cid:18) t V t (cid:18) Y ∂∂Z (cid:19) ∂∂ Ω (cid:19) are invariant under the action (1.2) of G J . The author [29] proved that for any twopositive real numbers A and B , the following differential operator(4.8) ∆ n,m ; A,B = 4 A M + 4 B M is the Laplacian of the G J -invariant Riemannian metric ds n,m ; A,B . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 17
Proposition 4.1.
The following differential operator K on H n,m of degree n definedby (4.9) K = det( Y ) det (cid:18) ∂∂Z t (cid:18) ∂∂Z (cid:19)(cid:19) is invariant under the action (1.2) of G J .Proof. Let K M, ( λ,µ ; κ ) denote the image of K under the transformation(Ω , Z ) (cid:0) ( M · Ω , ( Z + λ Ω + µ )( C Ω + D ) − (cid:1) with M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R ) and ( λ, µ ; κ ) ∈ H ( n,m ) R . If f is a C ∞ function on H n,m , using (4.4), (4.5) and Lemma 4.1, we have K M, ( λ,µ ; κ ) f = det( Y ) | det( C Ω + D ) | − det (cid:20) ( C Ω + D ) ∂∂Z t (cid:26) ( C Ω + D ) ∂f∂Z (cid:27)(cid:21) = det( Y ) | det( C Ω + D ) | − det (cid:20) ( C Ω + D ) t (cid:26) ( C Ω + D ) t (cid:18) ∂∂Z t (cid:18) ∂f∂Z (cid:19)(cid:19)(cid:27)(cid:21) = det( Y ) | det( C Ω + D ) | − det (cid:20) ( C Ω + D ) ∂∂Z t (cid:18) ∂f∂Z (cid:19) t ( C Ω + D ) (cid:21) = det( Y ) det (cid:18) ∂∂Z t (cid:18) ∂f∂Z (cid:19)(cid:19) = K f. Since M ∈ Sp ( n, R ) and ( λ, µ ; κ ) ∈ H ( n,m ) R are arbitrary, K is invariant under theaction (1.2) of G J . (cid:3) Proposition 4.2.
The following matrix-valued differential operator T on H n,m definedby (4.10) T = t (cid:18) ∂∂Z (cid:19) Y ∂∂Z is invariant under the action (1.2) of G J .Proof. Let T M, ( λ,µ ; κ ) denote the image of K under the transformation(Ω , Z ) (cid:0) ( M · Ω , ( Z + λ Ω + µ )( C Ω + D ) − (cid:1) with M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R ) and ( λ, µ ; κ ) ∈ H ( n,m ) R . If f is a C ∞ function on H n,m , according to (4.4), (4.5) and Lemma 4.1, we have T M, ( λ,µ ; κ ) f = t (cid:18) ( C Ω + D ) ∂∂Z (cid:19) t ( C Ω + D ) − Y ( C Ω + D ) − ( C Ω + D ) ∂f∂Z = t (cid:18) ∂∂Z (cid:19) Y ∂f∂Z = T f. Since M ∈ Sp ( n, R ) and ( λ, µ ; κ ) ∈ H ( n,m ) R are arbitrary, T is invariant under theaction (1.2) of G J . (cid:3) Corollary 4.1.
Each ( k, l ) -entry T kl of T given by (4.11) T kl = n X i,j =1 y ij ∂ ∂z ki ∂z lj , ≤ k, l ≤ m is an element of D (cid:0) H n,m (cid:1) .Proof. It follows immediately from Proposition 4.2. (cid:3)
Now we consider invariant differential operators on the Siegel-Jacobi disk. Let D n = (cid:8) W ∈ C ( n,n ) | W = t W, I n − W W > (cid:9) be the generalized unit disk.For brevity, we write D n,m := D n × C ( m,n ) . For a coordinate (
W, η ) ∈ D n,m with W = ( w µν ) ∈ D n and η = ( η kl ) ∈ C ( m,n ) , we put dW = ( dw µν ) , dW = ( dw µν ) ,dη = ( dη kl ) , dη = ( dη kl )and ∂∂W = (cid:18) δ µν ∂∂w µν (cid:19) , ∂∂W = (cid:18) δ µν ∂∂w µν (cid:19) ,∂∂η = ∂∂η . . . ∂∂η m ... . . . ... ∂∂η n . . . ∂∂η mn , ∂∂η = ∂∂η . . . ∂∂η m ... . . . ... ∂∂η n . . . ∂∂η mn . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 19
We can identify an element g = ( M, ( λ, µ ; κ )) of G J , M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R )with the element A B A t µ − B t λλ I m µ κC D C t µ − D t λ I m of Sp ( m + n, R ) . We set T ∗ = 1 √ (cid:18) I m + n I m + n iI m + n − iI m + n (cid:19) . We now consider the group G J ∗ defined by G J ∗ := T − ∗ G J T ∗ . If g = ( M, ( λ, µ ; κ )) ∈ G J with M = (cid:18) A BC D (cid:19) ∈ Sp ( n, R ), then T − ∗ gT ∗ is given by(4.12) T − ∗ gT ∗ = (cid:18) P ∗ Q ∗ Q ∗ P ∗ (cid:19) , where P ∗ = (cid:18) P { Q t ( λ + iµ ) − P t ( λ − iµ ) } ( λ + iµ ) I h + i κ (cid:19) ,Q ∗ = (cid:18) Q { P t ( λ − iµ ) − Q t ( λ + iµ ) } ( λ − iµ ) − i κ (cid:19) , and P, Q are given by the formulas(4.13) P = 12 { ( A + D ) + i ( B − C ) } and(4.14) Q = 12 { ( A − D ) − i ( B + C ) } . From now on, we write (cid:18)(cid:18)
P QQ P (cid:19) , (cid:18)
12 ( λ + iµ ) ,
12 ( λ − iµ ); − i κ (cid:19)(cid:19) := (cid:18) P ∗ Q ∗ Q ∗ P ∗ (cid:19) . In other words, we have the relation T − ∗ (cid:18)(cid:18) A BC D (cid:19) , ( λ, µ ; κ ) (cid:19) T ∗ = (cid:18)(cid:18) P QQ P (cid:19) , (cid:18)
12 ( λ + iµ ) ,
12 ( λ − iµ ); − i κ (cid:19)(cid:19) . Let H ( n,m ) C := (cid:8) ( ξ, η ; ζ ) | ξ, η ∈ C ( m,n ) , ζ ∈ C ( m,m ) , ζ + η t ξ symmetric (cid:9) be the complex Heisenberg group endowed with the following multiplication( ξ, η ; ζ ) ◦ ( ξ ′ , η ′ ; ζ ′ ) := ( ξ + ξ ′ , η + η ′ ; ζ + ζ ′ + ξ t η ′ − η t ξ ′ )) . We define the semidirect product SL (2 n, C ) ⋉ H ( n,m ) C endowed with the following multiplication (cid:18)(cid:18) P QR S (cid:19) , ( ξ, η ; ζ ) (cid:19) · (cid:18)(cid:18) P ′ Q ′ R ′ S ′ (cid:19) , ( ξ ′ , η ′ ; ζ ′ ) (cid:19) = (cid:18)(cid:18) P QR S (cid:19) (cid:18) P ′ Q ′ R ′ S ′ (cid:19) , ( ˜ ξ + ξ ′ , ˜ η + η ′ ; ζ + ζ ′ + ˜ ξ t η ′ − ˜ η t ξ ′ ) (cid:19) , where ˜ ξ = ξP ′ + ηR ′ and ˜ η = ξQ ′ + ηS ′ . If we identify H ( n,m ) R with the subgroup (cid:8) ( ξ, ξ ; iκ ) | ξ ∈ C ( m,n ) , κ ∈ R ( m,m ) (cid:9) of H ( n,m ) C , we have the following inclusion G J ∗ ⊂ SU ( n, n ) ⋉ H ( n,m ) R ⊂ SL (2 n, C ) ⋉ H ( n,m ) C . We define the mapping Θ : G J −→ G J ∗ by(4.15) Θ (cid:18)(cid:18) A BC D (cid:19) , ( λ, µ ; κ ) (cid:19) := (cid:18)(cid:18) P QQ P (cid:19) , (cid:18)
12 ( λ + iµ ) ,
12 ( λ − iµ ); − i κ (cid:19)(cid:19) , where P and Q are given by (4.13) and (4.14). We can see that if g , g ∈ G J , thenΘ( g g ) = Θ( g )Θ( g ) . According to [26, p. 250], G J ∗ is of the Harish-Chandra type (cf. [17, p. 118]). Let g ∗ = (cid:18)(cid:18) P QQ P (cid:19) , ( λ, µ ; κ ) (cid:19) be an element of G J ∗ . Since the Harish-Chandra decomposition of an element (cid:18)
P QR S (cid:19) in SU ( n, n ) is given by (cid:18) P QR S (cid:19) = (cid:18) I n QS − I n (cid:19) (cid:18) P − QS − R S (cid:19) (cid:18) I n S − R I n (cid:19) , the P + ∗ -component of the following element g ∗ · (cid:18)(cid:18) I n W I n (cid:19) , (0 , η ; 0) (cid:19) , W ∈ D n of SL (2 n, C ) ⋉ H ( n,m ) C is given by(4.16) (cid:18)(cid:18) I n ( P W + Q )( QW + P ) − I n (cid:19) , (cid:0) , ( η + λW + µ )( QW + P ) − ; 0 (cid:1)(cid:19) . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 21
We can identify D n,m with the subset (cid:26)(cid:18)(cid:18) I n W I n (cid:19) , (0 , η ; 0) (cid:19) (cid:12)(cid:12)(cid:12) W ∈ D n , η ∈ C ( m,n ) (cid:27) of the complexification of G J ∗ . Indeed, D n,m is embedded into P + ∗ given by P + ∗ = (cid:26) (cid:18)(cid:18) I n W I n (cid:19) , (0 , η ; 0) (cid:19) (cid:12)(cid:12)(cid:12) W = t W ∈ C ( n,n ) , η ∈ C ( m,n ) (cid:27) . This is a generalization of the Harish-Chandra embedding (cf. [17, p. 119]). Then weget the natural transitive action of G J ∗ on D n,m defined by (cid:18)(cid:18) P QQ P (cid:19) , (cid:0) ξ, ξ ; iκ (cid:1)(cid:19) · ( W, η )(4.17) = (cid:16) ( P W + Q )( QW + P ) − , ( η + ξW + ξ )( QW + P ) − (cid:17) , where (cid:18) P QQ P (cid:19) ∈ G ∗ , ξ ∈ C ( m,n ) , κ ∈ R ( m,m ) and ( W, η ) ∈ D n,m . The author [30] proved that the action (1.2) of G J on H n,m is compatible with theaction (4.17) of G J ∗ on D n,m through a partial Cayley transform Φ : D n,m −→ H n,m defined by(4.18) Φ( W, η ) := (cid:16) i ( I n + W )( I n − W ) − , i η ( I n − W ) − (cid:17) . In other words, if g ∈ G J and ( W, η ) ∈ D n,m ,(4.19) g · Φ( W, η ) = Φ( g ∗ · ( W, η )) , where g ∗ = T − ∗ g T ∗ . Φ is a biholomorphic mapping of D n,m onto H n,m which givesthe partially bounded realization of H n,m by D n,m . The inverse of Φ isΦ − (Ω , Z ) = (cid:16) (Ω − iI n )(Ω + iI n ) − , Z (Ω + iI n ) − (cid:17) . For (
W, η ) ∈ D n,m , we write (Ω , Z ) := Φ( W, η ) . Thus(4.20) Ω = i ( I n + W )( I n − W ) − , Z = 2 i η ( I n − W ) − . Since d ( I n − W ) − = ( I n − W ) − dW ( I n − W ) − and I n + ( I n + W )( I n − W ) − = 2 ( I n − W ) − , we get the following formulas from (4.20) Y = 12 i (Ω − Ω ) = ( I n − W ) − ( I n − W W )( I n − W ) − , (4.21) V = 12 i ( Z − Z ) = η ( I n − W ) − + η ( I n − W ) − , (4.22) d Ω = 2 i ( I n − W ) − dW ( I n − W ) − , (4.23) dZ = 2 i n dη + η ( I n − W ) − dW o ( I n − W ) − . (4.24)Using Formulas (4.18), (4.20)-(4.24), the author [31] proved that for any two posi-tive real numbers A and B , the following metric d ˜ s n,m ; A,B defined by ds D n,m ; A,B = 4 A tr (cid:16) ( I n − W W ) − dW ( I n − W W ) − dW (cid:17) + 4 B (cid:26) tr (cid:16) ( I n − W W ) − t ( dη ) β (cid:17) + tr (cid:16) ( ηW − η )( I n − W W ) − dW ( I n − W W ) − t ( dη ) (cid:17) + tr (cid:16) ( ηW − η )( I n − W W ) − dW ( I n − W W ) − t ( dη ) (cid:17) − tr (cid:16) ( I n − W W ) − t η η ( I n − W W ) − W dW ( I n − W W ) − dW (cid:17) − tr (cid:16) W ( I n − W W ) − t η η ( I n − W W ) − dW ( I n − W W ) − dW (cid:17) + tr (cid:16) ( I n − W W ) − t η η ( I n − W W ) − dW ( I n − W W ) − dW (cid:17) + tr (cid:16) ( I n − W ) − t η η W ( I n − W W ) − dW ( I n − W W ) − dW (cid:17) + tr (cid:16) ( I n − W ) − ( I n − W )( I n − W W ) − t η η ( I n − W W ) − × ( I n − W )( I n − W ) − dW ( I n − W W ) − dW (cid:17) − tr (cid:16) ( I n − W W ) − ( I n − W )( I n − W ) − t η η ( I n − W ) − × dW ( I n − W W ) − dW (cid:17)(cid:27) is a Riemannian metric on D n,m which is invariant under the action (4.17) of theJacobi group G J ∗ . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 23
We note that if n = m = 1 and A = B = 1 , we get14 ds D , ;1 , = dW dW (1 − | W | ) + 1(1 − | W | ) dη dη + (1 + | W | ) | η | − W η − W η (1 − | W | ) dW dW + ηW − η (1 − | W | ) dW dη + ηW − η (1 − | W | ) dW dη. From the formulas (4.20), (4.23) and (4.24), we get(4.25) ∂∂ Ω = 12 i ( I n − W ) (cid:20) t (cid:26) ( I n − W ) ∂∂W (cid:27) − t (cid:26) t η t (cid:18) ∂∂η (cid:19)(cid:27) (cid:21) and(4.26) ∂∂Z = 12 i ( I n − W ) ∂∂η . Using Formulas (4.20)-(4.22), (4.25), (4.26) and Lemma 4.1, the author [31] provedthat the following differential operators S and S on D n,m defined by S = σ (cid:18) ( I n − W W ) ∂∂η t (cid:18) ∂∂η (cid:19)(cid:19) and S = tr (cid:18) ( I n − W W ) t (cid:18) ( I n − W W ) ∂∂W (cid:19) ∂∂W (cid:19) + tr (cid:18) t ( η − η W ) t (cid:18) ∂∂η (cid:19) ( I n − W W ) ∂∂W (cid:19) + tr (cid:18) ( η − η W ) t (cid:18) ( I n − W W ) ∂∂W (cid:19) ∂∂η (cid:19) − tr (cid:18) ηW ( I n − W W ) − t η t (cid:18) ∂∂η (cid:19) ( I n − W W ) ∂∂η (cid:19) − tr (cid:18) ηW ( I n − W W ) − t η t (cid:18) ∂∂η (cid:19) ( I n − W W ) ∂∂η (cid:19) + tr (cid:18) η ( I n − W W ) − t η t (cid:18) ∂∂η (cid:19) ( I n − W W ) ∂∂η (cid:19) + tr (cid:18) η W W ( I n − W W ) − t η t (cid:18) ∂∂η (cid:19) ( I n − W W ) ∂∂η (cid:19) are invariant under the action (4.17) of G J ∗ . The author also proved that(4.27) ∆ D n,m ; A,B := 1 A S + 1 B S is the Laplacian of the invariant metric ds D n,m ; A,B on D n,m (cf. [31]). Proposition 4.3.
The following differential operator on D n,m defined by (4.28) K D = det( I n − W W ) det (cid:18) ∂∂η t (cid:18) ∂∂η (cid:19)(cid:19) is invariant under the action (4.17) of G J ∗ on D n,m .Proof. It follows from Proposition 4.1, Formulas (4.21), (4.26) and the fact that theaction (1.2) of G J on H n,m is compatible with the action (4 .
17) of G J ∗ on D n,m via thepartial Cayley transform. (cid:3) Proposition 4.4.
The following matrix-valued differential operator on D n,m definedby (4.29) T D := t (cid:18) ∂∂η (cid:19) ( I n − W W ) ∂∂η is invariant under the action (4.17) of G J ∗ on D n,m .Proof. It follows from Proposition 4.2, Formulas (4.21), (4.26) and the fact that theaction (1.2) of G J on H n,m is compatible with the action (4 .
17) of G J ∗ on D n,m via thepartial Cayley transform. (cid:3) Corollary 4.2.
Each ( k, l ) -entry T D kl of T D given by (4.30) T D kl = n X i,j =1 δ ij − n X r =1 w ir w jr ! ∂ ∂η ki ∂η lj , ≤ k, l ≤ m is a G J ∗ -invariant differential operator on D n,m .Proof. It follows immediately from Proposition 4.4. (cid:3)
For two differential operators D and D on H n,m or D n,m , we write[ D , D ] := D D − D D . Then(4.31) M = [ M , M ] = M M − M M is an invariant differential operator of degree three on H n,m and(4.32) P kl = [ K , T kl ] = KT kl − T kl K , ≤ k, l ≤ m is an invariant differential operator of degree 2 n + 1 on H n,m . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 25
Similarly(4.33) S = [ S , S ] = S S − S S is an invariant differential operator of degree three on D n,m and(4.34) Q kl = [ K D , T D kl ] = K D T D kl − T D kl K D , ≤ k, l ≤ m is an invariant differential operator of degree 2 n + 1 on D n,m .Indeed it is very complicated and difficult at this moment to express the generatorsof the algebra of all G J ∗ -invariant differential operators on D n,m explicitly.5. The Case n = m = 1 We consider the case n = m = 1 . For a coordinate ( ω, z ) in T , , we write ω = x + i y, z = u + i v, x, y, u, v real. The author [27] proved that the algebra Pol U (1)1 , isgenerated by q ( ω, z ) = 14 ω ω = 14 (cid:0) x + y (cid:1) ,ξ ( ω, z ) = z z = u + v ,φ ( ω, z ) = 12 Re (cid:0) z ω (cid:1) = 12 (cid:0) u − v (cid:1) x + uvy,ψ ( ω, z ) = 12 Im ( z ω ) = 12 (cid:0) v − u (cid:1) y + uvx. In [27], using Formula (3.11) the author calculated explicitly the images D = Θ , ( q ) , D = Θ , ( ξ ) , D = Θ , ( φ ) and D = Θ , ( ψ )of q, ξ, φ and ψ under the Halgason map Θ , . We can show that the algebra D ( H , )is generated by the following differential operators D = y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + v (cid:18) ∂ ∂u + ∂ ∂v (cid:19) + 2 y v (cid:18) ∂ ∂x∂u + ∂ ∂y∂v (cid:19) ,D = y (cid:18) ∂ ∂u + ∂ ∂v (cid:19) ,D = y ∂∂y (cid:18) ∂ ∂u − ∂ ∂v (cid:19) − y ∂ ∂x∂u∂v − (cid:18) v ∂∂v + 1 (cid:19) D and D = y ∂∂x (cid:18) ∂ ∂v − ∂ ∂u (cid:19) − y ∂ ∂y∂u∂v − v ∂∂u D , where τ = x + iy and z = u + iv with real variables x, y, u, v. Moreover, we have D D − D D = 2 y ∂∂y (cid:18) ∂ ∂u − ∂ ∂v (cid:19) − y ∂ ∂x∂u∂v − (cid:18) v ∂∂v D + D (cid:19) . In particular, the algebra D ( H , ) is not commutative. We refer to [1, 27] for moredetail.Recently Hiroyuki Ochiai [15] proved the following results. Theorem 5.1.
We have the following relation (5.1) φ + ψ = q ξ . This relation exhausts all the relations among the generators q, ξ, φ and ψ of Pol U (1)1 , . Theorem 5.2.
We have the following relations ( a ) [ D , D ] = 2 D ( b ) [ D , D ] = 2 D D − D ( c ) [ D , D ] = − D ( d ) [ D , D ] = 0( e ) [ D , D ] = 0( f ) [ D , D ] = 0( g ) D + D = D D D These seven relations exhaust all the relations among the generators D , D , D and D of D ( H , ) . We can prove the following
Theorem 5.3.
The action of U (1) on Pol U (1)1 , is not multiplicity-free. Finally we see that for the case n = m = 1, the seven problems proposed in Section3 are completely solved. NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 27
Remark 5.1.
According to Theorem 5.2, we see that D is a generator of the centerof D ( H , ) . We observe that the Lapalcian ∆ , A,B = 4
A D + 4 B D (see (4 . of ( H , , ds , A,B ) does not belong to the center of D ( H , ) . The Case n = 1 and m is arbitrary Conley and Raum [5] found the 2 m + m + 1 explicit generators of D ( H ,m ) andthe explicit one generator of the center of D ( H ,m ). They also found the generatorsof the center of the universal enveloping algebra of U (cid:0) g J (cid:1) of the Jacobi Lie algebra g J . The number of generators of the center of U (cid:0) g J (cid:1) is 1 + m ( m +1)2 . According to Theorem 3.2, Pol U (1)1 ,m is generated by q ( ω, z ) = tr( ω ω ) , (6.1) α kp ( ω, z ) = Re (cid:0) z t z (cid:1) kp = Re ( z k z p ) , ≤ k ≤ p ≤ m, (6.2) β lq ( ω, z ) = Im (cid:0) z t z (cid:1) lq = Im ( z l z q ) , ≤ l < q ≤ m, (6.3) f kp ( ω, z ) = Re ( z ω t z ) kp = Re ( ωz k z p ) , ≤ k ≤ p ≤ m, (6.4) g kp ( ω, z ) = Im ( z ω t z ) kp = Im ( ωz k z p ) , ≤ k ≤ p ≤ m, (6.5)where ω ∈ T and z ∈ C m .We let ω = x + iy ∈ C and z = t ( z , · · · , z m ) ∈ C m with z k = u k + iv k , ≤ k ≤ m, where x, y, u , v , · · · , u m , v m are real. The invariants q, α kp , β lq , f kp and g kp are ex-pressed in terms of x, y, u k , v l (1 ≤ k, l ≤ m ) as follows: q ( ω, z ) = x + y ,α kp ( ω, z ) = u k u p + v k v p , ≤ k ≤ p ≤ m,β lq ( ω, z ) = u q v l − u l v q , ≤ l < q ≤ m,f kp ( ω, z ) = x ( u k u p − v k v p ) + y ( u k v p + v k u p ) , ≤ k ≤ p ≤ m,g kp ( ω, z ) = x ( u k v p + v k u p ) − y ( u k u p − v k v p ) , ≤ k ≤ p ≤ m. Theorem 6.1.
The m ( m +1)2 relations (6.6) f kp + g kp = q α kk α pp , ≤ k ≤ p ≤ m exhaust all the relations among a set of generators q, α kp , β lq , f kp and g kp with ≤ k ≤ p ≤ m and ≤ l < q ≤ m . Theorem 6.2.
The action of U (1) on Pol ,m is not multiplicity-free. In fact, ifPol ,m = X σ ∈ [ U (1) m σ σ, then m σ = ∞ . Problem 1, Problem 2, Problem 4, Problem 5 and Problem 7 were solved. Problem3 can be handled. Finally Problem 6 is unsolved in the case that n = 1 and m isarbitrary. 7. Final Remarks
Using G J -invariant differential operators on the Siegel-Jacobi space, we introducea noton of Maass-Jacobi forms. Definition 7.1.
Let Γ n,m := Sp ( n, Z ) ⋉ H ( n,m ) Z be the discrete subgroup of G J , where H ( n,m ) Z = n ( λ, µ ; κ ) ∈ H ( n,m ) R | λ, µ, κ are integral o . A smooth function f : H n,m −→ C is called a Maass - Jacobi form on H n,m if f satisfiesthe following conditions (MJ1)-(MJ3) :(MJ1) f is invariant under Γ n,m . (MJ2) f is an eigenfunction of the Laplacian ∆ n,m ; A,B (cf. Formula (4.8)).(MJ3) f has a polynomial growth, that is, there exist a constant C > and apositive integer N such that | f ( X + iY, Z ) | ≤ C | p ( Y ) | N as det Y −→ ∞ , where p ( Y ) is a polynomial in Y = ( y ij ) . Remark 7.1.
Let D ∗ be a commutative subalgebra of D ( H n,m ) containing the Lapla-cian ∆ n,m ; A,B . We say that a smooth function f : H n,m −→ C is a Maass-Jacobiform with respect to D ∗ if f satisfies the conditions ( M J , ( M J ∗ and ( M J : thecondition ( M J ∗ is given by ( M J ∗ f is an eigenfunction of any invariant differential operator in D ∗ . It is natural to propose the following problems.
Problem A :
Find all the eigenfunctions of ∆ n,m ; A,B . Problem B :
Construct Maass-Jacobi forms.
NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 29
If we find a nice eigenfunction φ of the Laplacian ∆ n,m ; A,B , we can construct aMaass-Jacobi form f φ on H n,m in the usual way defined by(7.1) f φ (Ω , Z ) := X γ ∈ Γ ∞ n,m \ Γ n,m φ (cid:0) γ · (Ω , Z ) (cid:1) , where Γ ∞ n,m = (cid:26)(cid:18)(cid:18) A BC D (cid:19) , ( λ, µ ; κ ) (cid:19) ∈ Γ n,m (cid:12)(cid:12)(cid:12) C = 0 (cid:27) is a subgroup of Γ n,m . We consider the simple case n = m = 1 and A = B = 1. A metric ds , , on H , given by ds , , = y + v y ( dx + dy ) + 1 y ( du + dv ) − vy ( dx du + dy dv )is a G J -invariant K¨ahler metric on H , . Its Laplacian ∆ , , is given by∆ , , = y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + ( y + v ) (cid:18) ∂ ∂u + ∂ ∂v (cid:19) + 2 y v (cid:18) ∂ ∂x∂u + ∂ ∂y∂v (cid:19) . We provide some examples of eigenfunctions of ∆ , , .(1) h ( x, y ) = y K s − (2 π | a | y ) e πiax ( s ∈ C , a = 0 ) with eigenvalue s ( s − . Here K s ( z ) := 12 Z ∞ exp n − z t + t − ) o t s − dt, where Re z > . (2) y s , y s x, y s u ( s ∈ C ) with eigenvalue s ( s − . (3) y s v, y s uv, y s xv with eigenvalue s ( s + 1) . (4) x, y, u, v, xv, uv with eigenvalue 0.(5) All Maass wave forms.Let ρ be a rational representation of GL ( n, C ) on a finite dimensional complexvector space V ρ . Let M ∈ R ( m,m ) be a symmetric half-integral semi-positive definitematrix of degree m . Let C ∞ ( H n,m , V ρ ) be the algebra of all C ∞ functions on H n,m with values in V ρ . We define the | ρ, M -slash action of G J on C ∞ ( H n,m , V ρ ) as follows: If f ∈ C ∞ ( H n,m , V ρ ), f | ρ, M [( M, ( λ, µ ; κ ))](Ω , Z ):= e − πi tr( M [ Z + λ Ω+ µ ]( C Ω+ D ) − C ) · e πi tr( M ( λ Ω t λ + 2 λ t Z + κ + µ t λ )) (7.2) × ρ ( C Ω + D ) − f ( M · Ω , ( Z + λ Ω + µ )( C Ω + D ) − ) , where (cid:18) A BC D (cid:19) ∈ Sp ( n, R ) and ( λ, µ ; κ ) ∈ H ( n,m ) R . We recall the Siegel’s notation α [ β ] = t βαβ for suitable matrices α and β . We define D ρ, M to be the algebra of alldifferential operators D on H n,m satisfying the following condition(7.3) ( Df ) | ρ, M [ g ] = D ( f | ρ, M [ g ])for all f ∈ C ∞ ( H n,m , V ρ ) and for all g ∈ G J . We denote by Z ρ, M the center of D ρ, M .We define an another notion of Maass-Jacobi forms as follows. Definition 7.2.
A vector-valued smooth function φ : H n,m −→ V ρ is called a Maass-Jacobi form on H n,m of type ρ and index M if it satisfies the following conditions ( M J ρ, M , ( M J ρ, M and ( M J ρ, M : ( M J ρ, M φ | ρ, M [ γ ] = φ for all γ ∈ Γ n,m . ( M J ρ, M f is an eigenfunction of all differential operators in the center Z ρ, M of D ρ, M . ( M J ρ, M f has a growth condition φ (Ω , Z ) = O (cid:16) e a det Y · e π tr ( M [ V ] Y − ) (cid:17) as det Y −→ ∞ for some a > . The case n = 1 , m = 1 and ρ = det k ( k = 0 , , , · · · ) was studied by R. Bendtand R. Schmidt [1], A. Pitale [16] and K. Bringmann and O. Richter [3]. The case n = 1 , m =arbitrary and ρ = det k ( k = 1 , , · · · ) was dealt with by C. Conleyand M. Raum [5]. In [5] the authors proved that the center Z det k , M of D det k , M is thepolynomial algebra with one generator C k, M , the so-called Casimir operator which is a | det k , M -slash invariant differential operator of degree three. Bringmann and Richter [3]considered the Poincar´e series P k, M (the case n = m = 1) that is a harmonic Maass-Jacobi form in the sense of Definition 7.2 and investigated its Fourier expansion andits Fourier coefficients. Here the harmonicity of P k, M means that C k, M P k, M = 0,i.e., P k, M is an eigenfunction of C k, M with zero eigenvalue. Conley and Raum [5]generalized the results in [16] and [3] to the case n = 1 and m is arbitrary. Remark 7.2. In [2] , Bringmann, Conley and Richter proved that the center of thealgebra of differential operators invariant under the action of the Jacobi group over acomplex quadratic field is generated by two Casimir operators of degree three. Theyalso introduce an analogue of Kohnen’s plus space for modular forms of half-integralweight over K = Q ( i ) , and provide a lift from it to the space of Jacobi forms over K . NVARIANT DIFFERENTIAL OPERATORS ON SIEGEL-JACOBI SPACE 31
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Elements of the Representation Theory of the Jacobi Group , Progressin Mathematics, , Birkh¨auser, Basel, 1998.[2] K. Bringmann, C. Conley and O. K. Richter,
Jacobi forms over complex quadratic fields via thecubic Casimier operators , preprint.[3] K. Bringmann and O. K. Richter,
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