Let G be a real reductive Lie group with maximal compact subgroup K. Atiyah-Schmid showed that every discrete series representation of G can be realized as the index of a Dirac operator on the symmetric space G/K. Both Atiyah-Schmid and Connes-Moscovici computed the index and provided a geometric formula for the formal degree of discrete series representation. In this talk, I will discuss an alternative approach by studying the large time behavior of the Laplacian on G/K, which allows us go beyond the discrete series representation. The main tools we used is a geometric orbital integral formula obtained by Bismut. I will show how to detect the Plancherel measure for all tempered representations of G, which can be viewed as a generalization of the theorem of Atiyah-Schmid and Connes-Moscovici.