Rigid inner forms of real and $p$-adic groups

Abstract

We define a new cohomology set $H^1(u \to W,Z \to G)$ for an affine algebraic group $G$ and a finite central subgroup $Z$, both defined over a local field of characteristic zero, which is an enlargement of the usual first Galois cohomology set of $G$. We show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered $L$-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the case of real groups, we show that this description is correct using Shelstad’s work.