Nonconnected group GL(N) of the p-adic body, part 1

摘要

Let F be a nonarchimedean local field of characteristic zero, and let N be an even integer at least 2. Consider the following algebraic groups, both defined and split over F: G = GL(N) and H = SO(N + 1). Let G(+) = G x {1, theta}, where theta(2) = I and theta act on G as the nontrivial outer automorphism. It is a nonconnected group. Let (G) over tilde = G theta be the connected component that contains theta. The group R is an endoscopic group of G theta(+). A conjecture predicts that to every L-packet Pi(H) of admissible, irreducible, and tempered representations of H(F), we can associate an admissible and irreducible representation pi(+) of G(+)(F), so that the restriction to (G) over tilde (F) of the character of pi(+) is an endoscopic transfer of the character of Pi(H) (i.e., the sum of the characters of the representations belonging to Pi(H)). This notion of transfer is equivalent to simple equalities between characters of pi(+) and Pi(H). In the second part of this article, we give the construction that associates pi(+) to Pi(H), and we prove the equalities between their characters. We consider only L-packets of discrete series representations of "unipotent reduction." (This property includes representations that contain nonzero vectors invariant by an Iwahori subgroup.) Our result is conditional: we suppose a fundamental lemma for our pair (G(+), H). In the first part, we generalize for the group G+ certain results of harmonic analysis which are well known for connected groups. We also prove that the fundamental lemma for (G(+), H) implies a "nonstandard" fundamental lemma relying on the Lie algebras of the groups SO(N + 1) and Sp(N).