Harmonic Analysis on Homogeneous spaces and the Langlands program

Abstract: After the work of Harish-Chandra on L^2(G), the most natural class of spaces which one tries to understand are L^2(H\G), where H is a closed subgroup of G. Among these, the case when H is a fixed point of an involution has been rather well studied in the context of real groups. I will discuss how similar questions can be considered for p-adic groups where one does not understand the decomposition as well, and how one may hope to understand at least some cases using the Langlands program.