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

Abstract

For any Hermitian Lie group $G$ of tube type we give a geometric quantization procedure of certain $K_{\mathbb{C}}$-orbits in ${\mathfrak{p}}_{\mathbb{C}}^{\ast }$ to obtain all scalar type highest weight representations. Here $K_{\mathbb{C}}$ is the complexification of a maximal compact subgroup $K\subseteq G$ with corresponding Cartan decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ of the Lie algebra of $G$. We explicitly realize every such representation $\pi$ on a Fock space consisting of square integrable holomorphic functions on its associated variety $\mathrm{Ass}(\pi )\subseteq {\mathfrak{p}}_{\mathbb{C}}^{\ast }$. The associated variety $\mathrm{Ass}(\pi )$ is the closure of a single nilpotent $K_{\mathbb{C}}$-orbit ${\mathcal{O}}^{K_{\mathbb{C}}}\subseteq {\mathfrak{p}}_{\mathbb{C}}^{\ast }$ which corresponds by the Kostant–Sekiguchi correspondence to a nilpotent coadjoint $G$-orbit ${\mathcal{O}}^G\subseteq {\mathfrak{g}}^{\ast }$. The known Schrödinger model of $\pi$ is a realization on $L^2(\mathcal{O})$, where $\mathcal{O}\subseteq {\mathcal{O}}^G$ 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 $I$- and $K$-Bessel functions on Jordan algebras which appear in the measure of ${\mathcal{O}}^{K_{\mathbb{C}}}$, 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 $J$-Bessel function as well as explicit Whittaker vectors.