Jan Möllers

摘要

For any Hermitian Lie group $G$ of tube type we give a geometric quantization procedure of certain $K_CC$-orbits in $rakp_CC^*$ to obtain all scalar type highest weight representations. Here $K_CC$ is the complexification of a maximal compact subgroup $Ksubseteq G$ with corresponding Cartan decomposition $rakg=rakk+rakp$ 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 $Ass(pi)subseteqrakp_CC^*$. The associated variety $Ass(pi)$ is the closure of a single nilpotent $K_CC$-orbit $calO^{K_CC}subseteqrakp_CC^*$ which corresponds by the Kostant--Sekiguchi correspondence to a nilpotent coadjoint $G$-orbit $calO^Gsubseteqrakg^*$. The known Schrödinger model of $pi$ is a realization on $L^2(calO)$, where $calOsubseteqcalO^G$ 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 $calO^{K_CC}$, 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 $J$-Bessel function as well as explicit Whittaker vectors.