## 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 *K*_{C}-orbit *X* in *p*_{C} and the *L*^{2}-inner product involves a K-Bessel function as density. Here *K* \subseteq *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 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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© Toshiyuki Kobayashi