Analysis Seminar: Noncommutative Geometry Meets Harmonic Analysis on Reductive Symmetric Spaces

Abstract: A homogeneous space G/H is called a reductive symmetric space if G is a (real) reductive Lie group, and H is a symmetric subgroup of G, meaning that H is the subgroup fixed by some involution on G. The representation theory on reductive symmetric spaces was studied in depth in the 1990s by Erik van den Ban, Patrick Delorme, and Henrik Schlichtkrull, among many others. In particular, they obtained the Plancherel formula for the L^2 space of G/H. An important aspect is that this generalizes the group case, obtained by Harish-Chandra, which corresponds to the case when G = G' x G' and H is the diagonal subgroup.

In joint work with Alexandre Afgoustidis, Nigel Higson, and Peter Hochs, we are exploring this subject from the viewpoint of noncommutative geometry. In this talk, I will describe this new development with emphasis put on a concise and elementary overview of the harmonic analysis on reductive symmetric spaces, geared toward a general audience in the analysis group.

Host: Yanli Song