We prove Gaussian estimates for heat kernels on semisimple Lie groups by using the method of Bloch wave representation. We also give a large time asymptotic expansion for heat kernels on compact extensions of abelian Lie groups.
"On the large time behavior of heat kernels on Lie groups." Duke Math. J. 120 (2) 311 - 351, 1 November 2003. https://doi.org/10.1215/S0012-7094-03-12024-4