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