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

Masafumi Shimada

paper

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

arXiv:2603.00370

Abstract

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 $I ¨ _{t}^{G_{0}}$ associated with the Laplace--Beltrami operators $I^_{G_{0}}$ and $I^_{G}$ 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 $I ¨ _{t}^{G_{0}}$ and $I ¨ _{t}^{G}$ , we derive an integral formula relating the heat kernel $I ¨ _{t}^{G_{0}}$ to $I ¨ _{t}^{G}$ . In the special case of $G_{0} = S L (2, R)$ , we show that the integral formula of $I ¨ ^{S L (2, R)}$ is expressed in terms of the properties of Tchebycheff polynomials.

45 pages