Representation Theory Seminar: Alexander Shamov

Title: Sobolev globalizations of Harish-Chandra modules in real rank 1

Speaker: Alexander Shamov

Abstract: A representation of a real reductive group G in a complete topological vector space is, under reasonable assumptions, a topological completion of a Harish-Chandra module - its "algebraic skeleton" carrying only the action of a chosen maximal compact subgroup K and the Lie algebra g. In this talk I'm going to introduce an analog of Sobolev topologies on Harish-Chandra modules in the special case of groups G = SO_{1,n}. Completions with respect to these topologies are functorial and are, therefore, measuring intrinsic smoothness proprerties of vectors, independently of any functional models. On the other hand, for principal series, they reduce to classical Sobolev spaces on the real flag manifolds. Under certain natural assumptions, this characterizes the Sobolev globalizations uniquely.