  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.
