Discrete Series and Characters of the Component Group

Suppose G(R) is a real reductive group, with L-group ∨ G, and φ : WR → ∨ G is an L-homomorphism. There is a close relationship between the Lpacket associated to φ and characters the component group of the centralizer of the image of φ. This was originally observed by Shelstad [10], and further refined by Arthur, Kottwitz, Langlands and Shelstad. This is reinterpreted in [3], to (among other things) make it a canonical bijection. This involves a number of changes, including using the notion of strong real form, several strong real forms at once, and taking a cover of the component group. This cover is not necessarily a two-group, so the values of the character may not be just signs. An important special case is that of discrete series L-packets. It is worthwhile to make the correspondence between characters of (covers of) the centralizer and the (generalized) L-packet in this case. While this is a special case of [3], it isn’t so easy to extract it from there, or from the more elementary reference [1]. These notes consist of a mostly self-contained treatment of this special case, with some details in the case of unitary groups. We also address a question raised by Michael Harris about endoscopy. While the primary reference is The Langlands Classification and Irreducible Characters [3], for what is in this paper other more accessible references are sufficient. Lifting of Characters [1], which concentrates on the case of regular integral infinitesimal character has most of what is needed. Some of the approach here follows that of Algorithms for Representation Theory of Real Reductive Groups [4], for which Guide to the Atlas Software [2] is a good introduction.