Le groupe ${\bf GL}_{N}$ tordu, sur un corps $p$-adique, 2ème partie

Abstract

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 $F:$ ${\bf G}={\bf GL}_{N}$ and ${\bf H}={\bf SO}(N+1)$. Let ${\bf G}^+={\bf G}\rtimes \{1,\theta\}$, where $\theta^2=1$ and $\theta$ act on ${\bf G}$ as the nontrivial outer automorphism. It is a nonconnected group. Let $\tilde{{\bf G}}={\bf G}\theta$ be the connected component that contains $\theta$. We consider $L$-packets of admissible irreducible discrete series representations of ${\bf H}(F)$ of unipotent reduction. In a previous article with C. Moeglin [MW], we have described these $L$-packets. Let $\Pi^H$ be such an $L$-packet. We associate to $\Pi^H$ an admissible irreducible representation $\pi^+$ of ${\bf G}^+(F)$. We prove that the restriction ${\rm trace}_{\tilde{{\bf G}}}\,\pi^+$ to $\tilde{{\bf G}}(F)$ of the character of $\pi^+$ is a stable distribution. Granting a fundamental lemma for the pair (${\bf G}^+$, ${\bf H}$) to be true, we prove then that ${\rm trace}_{\tilde{{\bf G}}}\,\pi^+$ is an endoscopic transfer of the sum of the characters of the representations belonging to $\Pi^H$. The proof uses the Schneider-Stuhler theory to express ${\rm trace}_{\tilde{{\bf G}}}\,\pi^+$ as an orbital integral of a pseudocoefficient of $\pi^{+}$. We use the generalized Springer correspondence to compute this pseudocoefficient.