The heat kernel on $SL(2,\mathbb{R})$

TL;DR

This paper derives an explicit formula for the heat kernel on the noncompact semisimple Lie group SL(2,R) using spherical and Helgason-Fourier transforms, advancing understanding of heat propagation on such groups.

Contribution

It provides the first explicit formula for the heat kernel on SL(2,R), utilizing spherical and Helgason-Fourier transforms for homogeneous vector bundles.

Findings

Explicit heat kernel formula for SL(2,R)

Application of spherical and Helgason-Fourier transforms

Enhanced understanding of heat diffusion on noncompact Lie groups

Abstract

Let $G$ be a noncompact semisimple Lie group equipped with a certain invariant Riemannian metric. Then, we can consider a heat kernel function on $G$ associated to the Riemannian metric. We give an explicit formula for the heat kernel when $G = SL(2,\mathbb{R})$. The main tools are spherical transform and Helgason-Fourier transform for homogeneous vector bundles studied by R. Camporesi.