Orateur
Description
With Nigel Higson and Robert Yuncken, we have constructed a groupoid for a Riemannian symmetric space, or $G/K$ where $G$ is a real reductive group with maximal compact subgroup $K$, based on a general construction due to Omar. Geometrically, it is formed as a smooth family of homogeneous spaces $G/H$, where $H$ ranges over all maximal compact subgroups (conjugates of $K$) and their "limit groups", which in particular contain the unipotent part of a parabolic subgroup (so that a limit homogeneous space looks roughly like $G/N$, the domain relevant to the principal series). This is related to the so-called "boundary degenerations" that appear in the study of real spherical varieties (related to Harish-Chandra's theory of the constant term). The key advantage is that this nice geometric object has a (Lie) groupoid structure, and so has a corresponding $C^*$ algebra. The construction is aimed to assist in applying $C^*$-algebraic techniques to representation theory. While studying a groupoid instead of a group may at first seem to make the problem harder, in fact the groupoid turns out to be very simple both geometrically and algebraically, and this groupoid gets at the geometric core of Harish-Chandra's "philosophy of cusp forms". Similar results about limit groups are known for more general non-Riemannian spaces, where $K$ is replaced by an open subgroup of the fixed points of an involution. This suggests possible generalizations to this case; we shall discuss progress made in this direction near the end of the talk.