Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 24 (2014), No. 2, 373--396Copyright 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 moellers@imf.au.dk [Abstract-pdf] \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 [ Fulltext-pdf  (367  KB)] for subscribers only.