Self-dual Representations of Division Algebras and Weil Groups: a Contrast

If ρ is a selfdual representation of a group G on a vector space V over C, we will say that ρ is orthogonal, resp. symplectic, if G leaves a nondegenerate symmetric, resp. alternating, bilinear form B : V ×V → C invariant. If ρ is irreducible, exactly one of these possibilities will occur, and we may define a sign c(ρ) ∈ {±1}, taken to be +1, resp. −1, in the orthogonal, resp. symplectic, case. Now let k be a local field of characteristic 0. The groups of interest to us will be G = GLm(D), where D is a division algebra with center k and index d, and the Weil group Wk. The local Langlands correspondence ([HT], [Hen1]) when used in conjunction with the Jacquet-Langlands correspondence ([Bdl]), gives a bijection π → σ, satisfying certain natural properties, in particular the preservation of ε-factors of pairs, between the discrete series representations of G and the set of irreducible representations σ of W ′ k of dimension n = md. Here W ′ k denotes Wk if k is Archimedean, and the extended Weil group Wk × SL2(C) if k is non-Archimedean. One calls σ the Langlands parameter of π. It is immediate from the construction that π is selfdual if and only if σ is. However, the local Langlands reciprocity is not a priori sensitive to the finer question of whether c(π) equals c(σ) or −c(σ). The main result of this paper is the following.