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

For any Hermitian Lie group G of tube type we give a geometric quantization procedure of certain KC -orbits in pC^* to obtain all scalar type highest weight representations. Here KC is the complexification of a maximal compact subgroup K⊆ G with corresponding Cartan decomposition g= k+ p of the Lie algebra of G . We explicitly realize every such representation π on a Fock space consisting of square integrable holomorphic functions on its associated variety Ass(π)⊆ pC^* . The associated variety Ass(π) is the closure of a single nilpotent KC -orbit O^{KC}⊆ pC^* which corresponds by the Kostant–Sekiguchi correspondence to a nilpotent coadjoint G -orbit O^G⊆ g^* . The known Schrödinger model of π is a realization on L^2( O) , where O⊆ 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 O^{KC} , 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.