C*-blocks and crossed products for real and p-adic reductive groups

Let G be a real or a p-adic connected reductive group. We will recall the description of the connected components of the tempered dual of G in terms of certain subalgebras of its reduced C*-algebra.

Each connected component comes with a torus equipped with a finite group action. We will see that, under a certain geometric assumption on the structure of stabilizers for that action (that is always satisfied for real groups), the component has a simple geometric structure which encodes the reducibility of the associate parabolically induced representations.

We will provide a characterization of the connected components for which the geometric assumption is satisfied, in the case when G is a symplectic group.

This is a joint work with Alexandre Afgoustidis.