A geometric quantization of the Kostant-Sekiguchi correspondence for scalar type unitary highest weight representations (Q374016)

Summary: For any Hermitian Lie group \(G\) of tube type we give a geometric quantization procedure of certain \(K_{\mathbb C}\)-orbits in \(\mathfrak p_{\mathbb C}^*\) 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 \(\text{Ass}(\pi)\subseteq\mathfrak p_{\mathbb C}^*\). The associated variety \(\text{Ass}(\pi)\) is the closure of a single nilpotent \(K_{\mathbb C}\)-orbit \(\mathcal O^{K_{\mathbb C}}\subseteq\mathfrak p_{\mathbb C}^*\) which corresponds by the Kostant-Sekiguchi correspondence to a nilpotent coadjoint \(G\)-orbit \(\mathcal O^G\subseteq\mathfrak g^*\). 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.