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

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 ${\rho }_t^{G_0}$ associated with the Laplace–Beltrami operators ${\mathrm{\Delta }}_{G_0}$ and ${\mathrm{\Delta }}_G$ 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 ${\rho }_t^{G_0}$ and ${\rho }_t^G$, we derive an integral formula relating the heat kernel ${\rho }_t^{G_0}$ to ${\rho }_t^G$. 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.