
Journal of Lie Theory 24 (2014), No. 2, 373396 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 moellers@imf.au.dk [Abstractpdf] \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 oneparameter family of elliptic selfadjoint 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, SegalBargmann transform, symmetric cone, Bergman space, Bessel operator, Bessel function. MSC: 58J35; 22E45, 30H20, 33C70 [ Fulltextpdf (367 KB)] for subscribers only. 