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

Masafumi Shimada

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 $ρ^{G_0}_t$ associated with the Laplace--Beltrami operators $Δ_{G_0}$ and $Δ_{G}$ 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 $ρ^{G_0}_t$ and $ρ^{G}_t$, we derive an integral formula relating the heat kernel $ρ^{G_0}_t$ to $ρ^{G}_t$. In the special case of $G_0=SL(2,\mathbb{R})$, we show that the integral formula of $ρ^{SL(2,\mathbb{R})}$ is expressed in terms of the properties of Tchebycheff polynomials.

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

Abstract

Let be a connected semisimple Lie group, and be its connected split real form. In this paper, we deduce explicit expressions for the heat kernels associated with the Laplace--Beltrami operators and 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 and , we derive an integral formula relating the heat kernel to . In the special case of , we show that the integral formula of is expressed in terms of the properties of Tchebycheff polynomials.
Paper Structure (19 sections, 80 equations)

This paper contains 19 sections, 80 equations.

Table of Contents

  1. Introduction
  2. The focus of this paper and main problems
  3. Background
  4. Overview of main results
  5. Summary of main results
  6. The organization
  7. Heat kernels and heat Gaussians
  8. The heat kernel on a Lie group
  9. The heat Gaussian on a Riemnnain symmetric space
  10. An explicit formula for the heat kernel on a semisimple Lie group
  11. Invariant differential operators on a homogeneous space
  12. Explicit formulae for $\rho_t^{G_0}$ and $\rho_t^G$
  13. A solution to the heat equation
  14. The initial condition
  15. An integral expression of the heat kernel on a real group
  16. ...and 4 more sections