Heat kernel analysis for Bessel operators on symmetric cones

Abstract

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 Bergmann space on the complexification $G_{\mathbb C}/K_{\mathbb C}$ 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 $\Omega=\mathbb{R}_+$ these results seem to be new.