Let G be a real or a p-adic connected reductive group. We will consider the connected components of the tempered dual of G. They are labelled by the G-conjugacy classes of pairs formed by a Levi subgroup M of G and the orbit of a discrete series representation of M under the group of unitary unramified characters of M.
For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple geometric structure which encodes the reducibility of the corresponding parabolically induced representations. We will explain how one can recover his result.
For p-adic groups, each connected component comes with a compact torus equipped with a finite group action, and an analogous result, that we will describe, holds true under a certain geometric assumption on the structure of stabilizers for that action. In the case when G is a quasi-split symplectic, special orthogonal or unitary group, it is possible to explicitly determine the connected components for which the geometric assumption is satisfied.
It is a joint work with Alexandre Afgoustidis.