Fock model and Segal–Bargmann transform for minimal representations of Hermitian Lie groups

Abstract

For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent KC-orbit X in pC and the L2-inner product involves a K-Bessel function as density. Here K⊆G is a maximal compact subgroup and gC=kC+pC is a complexified Cartan decomposition. In this realization the space of k-finite vectors consists of holomorphic polynomials on X. The reproducing kernel of the Fock space is calculated explicitly in terms of an I-Bessel function. We further find an explicit formula of a generalized Segal–Bargmann transform which intertwines the Schrödinger and Fock model. Its kernel involves the same I-Bessel function. Using the Segal–Bargmann transform we also determine the integral kernel of the unitary inversion operator in the Schrödinger model which is given by a J-Bessel function.