Hypoelliptic Laplacian and twisted trace formula

TL;DR

This paper presents a geometric formula for twisted orbital integrals using hypoelliptic Laplacian methods, enabling evaluation of heat operator traces and revisiting index theorems and torsions on symmetric spaces.

Contribution

It introduces an explicit geometric formula for twisted orbital integrals via hypoelliptic Laplacian, advancing the analysis of heat traces and index theorems on symmetric spaces.

Findings

Derived explicit formulas for twisted orbital integrals.

Evaluated equivariant heat trace using the new formula.

Revisited local index theorems and L2-torsions in this context.

Abstract

We give an explicit geometric formula for the twisted orbital integrals using the method of the hypoelliptic Laplacian developed by Bismut. Combining with the twisted trace formula, we can evaluate the equivariant trace of the heat operators of the Laplacians on a compact locally symmetric space. In particular, we revisit the equivariant local index theorems and twisted $L_{2}$-torsions for locally symmetric spaces.