$C^*$-blocks and crossed products for classical $p$-adic groups

Abstract: Let $G$ be the group of $F$-points of a quasi-split reductive connected group over a local field $F$. For $F$ real, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component of the tempered dual of $G$ has a simple and beautiful geometric structure that encodes the reducibility of induced representations. For $F$ $p$-adic, the existence of such a structure is far from straightforward. It was established for certain particular connected components by R. Plymen and his students.

We will present a joint work with Alexandre Afgoustidis which first provides a necessary and sufficient condition, in terms of the Knapp-Stein-Silberger R-groups, for the existence of a Wassermann type theorem, and secondly determine explicitly the components for which this condition is satisfied when $G$ is a classical $p$-adic group.

number theoryrepresentation theory

Audience: researchers in the topic

Number Theory and Representations in Valparaiso