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

TL;DR

This paper constructs a Fock model for minimal representations of Hermitian Lie groups of tube type, explicitly computes the reproducing kernel, and derives a Segal-Bargmann transform linking Schrödinger and Fock models.

Contribution

It introduces a new Fock space realization for minimal representations of Hermitian Lie groups of tube type with explicit kernel formulas and a Segal-Bargmann transform.

Findings

Explicit Fock space construction with K-Bessel density

Reproducing kernel expressed via I-Bessel function

Segal-Bargmann transform intertwines models and involves I-Bessel function

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 K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as density. Here K is a maximal compact subgroup of G, 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 Schroedinger 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 Schroedinger model which is given by a J-Bessel function.