Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

Abstract

Abstract. We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.

The Real Jacobi Group Revisited

“…We have determined the invariant metric on the Siegel-Jacobi upper half-plane X J 1 from the metric on D J 1 and the FC-transforms, see [13,14,18,21]. For the actions in Proposition 2.1, where G J 0 = SU(1, 1) C, see [17, Proposition 2] and Lemma 5.1 below.…”

“…Our special interest to the Jacobi group comes from the fact that G J n is a coherent state (CS) group [79,80,84,85,86,87], i.e., a group which has orbits holomorphically embedded into a projective Hilbert space, for a precise definition see [12,Definition 1], [13], [22,Section 5.2.2] and [29,Remark 4.4]. To an element X in the Lie algebra g of G we associated a first order differential operator X on the homogenous space G/H, with polynomial holomorphic coefficients, see [23,24,25] for CS based on hermitian symmetric spaces, where the maximum degree of the polynomial is 2.…”

“…It was proved in [16,17,21] that the Kähler two-form on D J n , invariant to the action of the Jacobi group G J n , has the expression −iω D J n (z, W ) = k 2 Tr(B ∧B) + µ Tr A tM ∧Ā , A = dz + dWη,…”

“…The properties of geodesics on the Siegel-Jacobi disk D J 1 have been investigated in [13,19,20], while in [22] we have considered geodesics on the Siegel-Jacobi ball D J n . We have explicitly determined the equations of geodesics on D J n .…”

Adjoint orbits of the Jacobi group

“…Therefore the Jacobi group G J plays an important role in number theory (e.g. theory of Jacobi forms) [4,13,14,15,16,17,18,27,29,32,33], algebraic geometry [20,22,29,31], complex geometry [23,24,25,26,29], representation theory [2,28,30] and mathematical physics [1,8].…”