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 $\rho^{G_0}_t$ 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.