Abstract: For a real reductive group G, an important problem in harmonic analysis is to understand the image of various convolution algebras under Fourier transform. This also provides insight to the representation theory of G. For the reduced group C*-algebra, the structure of this image was laid out cleanly by Clare, Crisp and Higson, and as a consequence it summaries central results in the theory of tempered representations, as well as allowing one to "read off" the topological structure of the tempered dual. For compactly supported smooth functions, the problem of determining this image is known as the Paley-Wiener problem, and says something about the dual of admissible representations of G. On the algebraic side, the Hecke algebra (K-finite distributions of G supported in K) is tied to the category of (g, K)-modules, the algebraic analogue of G-representations. When one computes this Fourier image for SL(2, R), one quickly finds that these function spaces look like matrix algebras of even functions multiplied by fixed polynomials. That is, the space is free over even functions. For example, the Hecke algebra for SL(2, R) is free over the center of the enveloping algebra, and one has analogous results for the other mentioned convolution algebras. In this talk, we will explore this point and try to develop freeness properties for more general groups. If there is time, I will explain its relevance to me in a pursuit of an "Oka principle" approach to the Baum-Connes conjecture. This talk is essentially an advertisement of a 1997 paper by Bernstein, Braverman and Gaitsgory regarding Cohen-Macaulay properties of the category of (g, K)-modules.
Bei Interesse an einer online-Teilnahme (sei es regelmäßig oder auch nur an einem bestimmten Vortrag) bitten wir vorab mit Tobias Weich oder Benjamin Delarue Kontakt aufzunehmen, damit der Teilnahmelink geteilt werden kann.