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Journal of Lie Theory 24 (2014), No. 2, 373--396
Copyright Heldermann Verlag 2014

Heat Kernel Analysis for Bessel Operators on Symmetric Cones

Jan Möllers
Institut for Matematiske Fag, Aarhus Universitet, Ny Munkegade 118, 8000 Aarhus C, Danmark


\def\C{{\Bbb C}} \def\R{{\Bbb R}} We investigate the heat equation corresponding to the Bessel operators on a symmetric cone $\Omega=G/K$. These operators form a one-parameter family of elliptic self-adjoint second order differential operators and occur in the Lie algebra action of certain unitary highest weight representations. The heat kernel is explicitly given in terms of a multivariable $I$-Bessel function on $\Omega$. Its corresponding heat kernel transform defines a continuous linear operator between $L^p$-spaces. The unitary image of the $L^2$-space under the heat kernel transform is characterized as a weighted Bergman space on the complexification $G_\C/K_\C$ of $\Omega$, the weight being expressed explicitly in terms of a multivariable $K$-Bessel function on $\Omega$. Even in the special case of the symmetric cone $\Omega=\R_+$ these results seem to be new.

Keywords: Heat kernel transform, Segal-Bargmann transform, symmetric cone, Bergman space, Bessel operator, Bessel function.

MSC: 58J35; 22E45, 30H20, 33C70

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