  Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual
of $G$, and its connected components. 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 induced
representations. For $p$-adic groups, each connected component of the tempered
dual comes with a compact torus equipped with a finite group action, and we
prove that a version of Wassermann's theorem holds true under a certain
geometric assumption on the structure of stabilizers for that action. We then
focus on the case where $G$ is a quasi-split symplectic, orthogonal or unitary
group, and explicitly determine the connected components for which the
geometric assumption is satisfied.
