Appears in collection : 2025 - T1 - WS2 - Tempered representations and K-theory

Let $G$ be a connected linear real reductive group with a maximal compact subgroup $K$. In this talk, we will explain an approach to study the large-time behavior of the heat kernel on the corresponding homogeneous space $G/K$ using Bismut’s formula. We will discuss how Bismut’s formula provides a natural link between the index theory and Vogan's minimal $K$-type theory. In particular, we will show that Vogan's $\lambda$-map plays a central role in the large time asymptotics analysis about the trace of the heat kernel. This talk is based on joint works with Shu Shen and Yanli Song.