Discrete Series and Characters of the Component Group

Suppose φ: WR → L G is an L-homomorphism. There is a close relationship between the L-packet associated to φ and characters the component group of the centralizer of φ. This is reinterpreted in [2], see also [1], in part to make it more canonical and a 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. Over the years a number of people have asked me how to relate an Lpacket of discrete series to characters of Sφ in this language. While this is a special case of [2] and [1], it isn’t so easy to extract it. In these notes I work out this case, and give some details in the case of SU(p, q) and U(p, q). There are no proofs; see the references for more details. Besides the basic references cited above, I make some use of [?]. If you want to cut to the chase the main result is Proposition 4.4. Also see Propositions 3.3, 7.3 and 8.2. 1 Basic Setup We fix a pair (G, γ) consisting of a reductive, algebraic (or complex) group and an outer automorphism of G. This corresponds to an inner class of real forms of G. We are interested in discrete series representations of real forms of G. A real form in this inner class has discrete series representations if and only if γ = 1, so we assume this. The extended group is a semidirect product G Γ = G ⋊ Γ where Γ = Gal(C/R). Since γ = 1 it is a direct product, and we may safely drop Γ from 1 the notation. The constructions of [2],[1],[?] involving the “twist ” simplify in this setting. We fix Cartan and Borel subgroups H and B of G. Let G ∨ be the dual group, and fix Cartan and Borel subgroups H ∨ and B ∨ of G ∨. By construction we have canonical identifications (1.1) X ∗ (H) = X∗(H ∨), X∗(H) = X ∗ (H ∨) where X ∗ and X ∗ denote the character and co-character lattices, respectively