# The Heat Operator of a Transversally Elliptic Operator

## TL;DR

This paper investigates the heat operator associated with a $G$-transversally elliptic operator on a compact Lie group, defining a character as a distribution that generalizes the heat trace in the equivariant setting.

## Contribution

It introduces a new character distribution for $G$-transversally elliptic operators and provides estimates for its convergence, extending spectral analysis in equivariant elliptic theory.

## Key findings

- Defined the character as a distribution on $G$
- Provided estimates for the convergence of the character
- Extended spectral properties to the equivariant case

## Abstract

Let $G$ be a connected compact Lie group. We study the heat operator of a $G$-transversally elliptic operator. After we review the spectral properties of a $G$-transversally elliptic operator, we define the character, that is a distribution on $G$ generalizing the trace of the heat operator to the $G$-equivariant case. The main theorem of this paper gives the estimate of $f_\alpha(t)$, which essentially determines the convergence of the character.