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

摘要

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 L ~2-inner product involves a K-Bessel function as density. Here K?G is a maximal compact subgroup and g _C=k _C+p _C 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 Schr?dinger 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 Schr?dinger model which is given by a J-Bessel function.