In 1980s, Connes and Moscovici studied index theory of G-invariant elliptic pseudo-differential operators acting on non-compact homogeneous spaces. They proved a L2 -index formula using the heat kernel method, which is related to the discrete series representation of Lie groups. In this talk, I will discuss the orbital integral of heat kernel and its relation with Plancherel formula. This is a generalization of the analytic index studied by Connes-Moscovici to the limit of discrete series case. In a recent work by Hochs-Wang, they obained a fixed point theorem for the topogical side of the index.