The heat kernel on a complex semisimple Lie group and an integral presentation of the heat kernel on its split real form

TL;DR

This paper derives explicit formulas for heat kernels on complex semisimple Lie groups and their split real forms, and establishes an integral relation between these kernels, with special cases involving Tchebycheff polynomials.

Contribution

It provides new explicit expressions for heat kernels on semisimple Lie groups and their split real forms, and introduces an integral formula linking these kernels, including a special case analysis for SL(2,R).

Findings

Explicit heat kernel formulas involving Gaussian and maximal compact subgroup kernels.

An integral relation connecting heat kernels of G and G_0.

Special case for SL(2,R) using Tchebycheff polynomials.

Abstract

Let $G$ be a connected semisimple Lie group, and $G_0$ be its connected split real form. In this paper, we deduce explicit expressions for the heat kernels $\rho^{G_0}_t$ associated with the Laplace--Beltrami operators $\Delta_{G_0}$ and $\Delta_{G}$ respectively, using the algebra of differential operators on an appropriate homogeneous space. These expressions involve the heat Gaussian and the heat kernel on a maximal compact subgroup. Using these expressions for $\rho^{G_0}_t$ and $\rho^{G}_t$, we derive an integral formula relating the heat kernel $\rho^{G_0}_t$ to $\rho^{G}_t$. In the special case of $G_0=SL(2,\mathbb{R})$, we show that the integral formula of $\rho^{SL(2,\mathbb{R})}$ is expressed in terms of the properties of Tchebycheff polynomials.