Integral representation of the subelliptic heat kernel on the complex anti-de Sitter fibration

TL;DR

This paper presents a novel integral representation for the subelliptic heat kernel on the complex anti-de Sitter fibration, highlighting new connections with the generalized Maass Laplacian and hyperbolic geometry.

Contribution

It introduces a different proof method based on commutativity properties, offering new insights into the sub-Laplacian and its relation to hyperbolic spaces.

Findings

Derived an integral formula for the heat kernel

Connected the sub-Laplacian to the generalized Maass Laplacian

Highlighted the role of odd-dimensional hyperbolic space

Abstract

We derive an integral representation for the subelliptic heat kernel of the complex anti-de Sitter fibration. Our proof is different from the one used in Jing Wang \cite{Wan} since it appeals to the commutativity of the D'Alembertian and of the Laplacian acting on the vertical variable rather than the analytic continuation of the heat semigroup of the real hyperbolic space. Our approach also sheds the light on the connection between the sub-Laplacian of the above fibration and the so-called generalized Maass Laplacian, and on the role played by the odd dimensional real hyperbolic space.