Equivariant heat asymptotics on spaces of automorphic forms

TL;DR

This paper derives precise asymptotic formulas for heat traces on compact symmetric spaces with symmetry group actions, including explicit leading coefficients for heat kernel expansions on homogeneous vector bundles.

Contribution

It provides the first detailed $K$-equivariant heat trace asymptotics with remainder estimates on symmetric spaces of arbitrary rank.

Findings

Computed the leading coefficient in the Minakshishundaram-Pleijel expansion.

Established $K$-equivariant asymptotics with explicit remainder estimates.

Extended heat trace asymptotics to general homogeneous vector bundles.

Abstract

Let $G$ be a connected, real, semisimple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. In this paper, we derive $K$-equivariant asymptotics for heat traces with remainder estimates on compact Riemannian manifolds carrying a transitive and isometric $G$-action. In particular, we compute the leading coefficient in the Minakshishundaram-Pleijel expansion of the heat trace for Bochner-Laplace operators on homogeneous vector bundles over compact locally symmetric spaces of arbitrary rank.