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 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.
Keywords
Cite
@article{arxiv.1203.5462,
title = {Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups},
author = {Joachim Hilgert and Toshiyuki Kobayashi and Jan Möllers and Bent Ørsted},
journal= {arXiv preprint arXiv:1203.5462},
year = {2014}
}
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77pages