J. Hilgert, T. Kobayashi, J. Möllers, and B. Ørsted, Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups, Journal of Functional Analysis 263 (2012), 3492-3563. DOI: 10.1016/j.jfa.2012.08.026. arXiv:1203.5462..

For any Hermitian Lie group G of tube type we construct a Fock model of its . 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 \subseteq 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.

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