Heat kernel, large-time behavior, and representation theory

Abstract

Given a real reductive group $G$, the purpose of this paper is to show an asymptotic formula of the large-time behavior of the $G$-trace of the heat operator on the associated symmetric spaces. Together with Carmona's proof on Vogan's lambda map, our results provide a geometric counterpart of Vogan's minimal $K$-type theory.

50 pages; the computation of the Novikov-Shubin type invariant for smooth locally symmetric spaces is added