On parameters for the group SO(2n)

This article is an attempt to refine a property of automorphic representations from the monograph [A3]. It concerns the local and global representations of an even, quasisplit, special orthogonal group (1.1) G = SO(N), N = 2n, over a field F of characteristic 0. The methods of [A3] are by comparison with representations of the group GL(N). This eventually leads to results on the rep-resentations of the full orthogonal group O(N). But it is the connected subgroup G = SO(N) of O(N) that is the ultimate object of interest. In [A3, §8.4], we were able to characterize certain representations of G from the results obtained earlier for O(N). The representations are those associated to Langlands parameters φ ∈ Φbdd(G). In the case of local F, Theorem 8.4.1 of [A3] provides an endoscopic classification of the representations of G(F). For global F, however, the associated automorphic representations are governed by a larger class of parameters ψ ∈ Ψ(G). An understanding of their local components requires supplementary endoscopic character relations for the localizations ψv ∈ Ψ(Gv) of ψ. In this article, we shall establish conditional analogues for ψv of the results for φv in [A3, §8.4]. The conditions we impose are local, and include properties established for p-adic F by Moeglin. Such properties seem to be of considerable interest in their own right, as we will try to indicate with a few supplementary remarks in §3. In §4, we will use the conditions to formulate a conjecture on the contribution of a global parameter ψ to the automorphic discrete spectrum of G. We will then sketch a proof of the local results in §5. Until further notice, the field F will be local. We then have the local Langlands group LF = WF, if F is archimedean, WF × SU(2), if F is p-adic, where WF is the local Weil group of F. Let us recall the basic objects from [A3] that we will be working with here