A geometric quantization of the Kostant-Sekiguchi correspondence for scalar type unitary highest weight representations

  • Jan Möllers

A geometric quantization of the Kostant-Sekiguchi correspondence for scalar type unitary highest weight representations cover
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Abstract

For any Hermitian Lie group GG of tube type we give a geometric quantization procedure of certain K\CCK_\CC-orbits in \frakp\CC\frakp_\CC^* to obtain all scalar type highest weight representations. Here K\CCK_\CC is the complexification of a maximal compact subgroup KGK\subseteq G with corresponding Cartan decomposition \frakg=\frakk+\frakp\frakg=\frakk+\frakp of the Lie algebra of GG. We explicitly realize every such representation π\pi on a Fock space consisting of square integrable holomorphic functions on its associated variety \Ass(π)\frakp\CC\Ass(\pi)\subseteq\frakp_\CC^*. The associated variety \Ass(π)\Ass(\pi) is the closure of a single nilpotent K\CCK_\CC-orbit \calOK\CC\frakp\CC\calO^{K_\CC}\subseteq\frakp_\CC^* which corresponds by the Kostant--Sekiguchi correspondence to a nilpotent coadjoint GG-orbit \calOG\frakg\calO^G\subseteq\frakg^*. The known Schrödinger model of π\pi is a realization on L2(\calO)L^2(\calO), where \calO\calOG\calO\subseteq\calO^G is a Lagrangian submanifold. We construct an intertwining operator from the Schrödinger model to the new Fock model, the generalized Segal--Bargmann transform, which gives a geometric quantization of the Kostant--Sekiguchi correspondence (a notion invented by Hilgert, Kobayashi, Ørsted and the author). The main tool in our construction are multivariable II- and KK-Bessel functions on Jordan algebras which appear in the measure of \calOK\CC\calO^{K_\CC}, as reproducing kernel of the Fock space and as integral kernel of the Segal--Bargmann transform. As a corollary to our construction we also obtain the integral kernel of the unitary inversion operator in the Schrödinger model in terms of a multivariable JJ-Bessel function as well as explicit Whittaker vectors.

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Jan Möllers, A geometric quantization of the Kostant-Sekiguchi correspondence for scalar type unitary highest weight representations. Doc. Math. 18 (2013), pp. 785–855

DOI 10.4171/DM/414