A geometric quantization of the Kostant-Sekiguchi correspondence for scalar type unitary highest weight representations
For any Hermitian Lie group of tube type we give a geometric quantization procedure of certain -orbits in to obtain all scalar type highest weight representations. Here is the complexification of a maximal compact subgroup with corresponding Cartan decomposition of the Lie algebra of . We explicitly realize every such representation on a Fock space consisting of square integrable holomorphic functions on its associated variety . The associated variety is the closure of a single nilpotent -orbit which corresponds by the Kostant--Sekiguchi correspondence to a nilpotent coadjoint -orbit . The known Schrödinger model of is a realization on , where 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 - and -Bessel functions on Jordan algebras which appear in the measure of , 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 -Bessel function as well as explicit Whittaker vectors.
Cite this article
Jan Möllers, A geometric quantization of the Kostant-Sekiguchi correspondence for scalar type unitary highest weight representations. Doc. Math. 18 (2013), pp. 785–855DOI 10.4171/DM/414