Euler-Poincaré pairings and elliptic representations of Weyl groups and \(p\)-adic groups. (Q2767767)

Let \(F\) be a nonarchimedean local field. Let \(G\) be a connected split adjoint reductive group over \(F\). The main result here is that elliptic representations of \(G(F)\) with Iwahori fixed vectors are isometrically related to elliptic representations of the Weyl groups of endoscopic groups of \(G\). \textit{D. Kazhdan} [J. Anal. Math. 47, 1--36 (1986; Zbl 0634.22009)], defined the space of elliptic representations of \(G(F)\) and an inner product on it by means of Harish-Chandra characters, when char \(F\) is 0. This pairing can be defined homologically for any characteristic, see [\textit{P. Schneider} and \textit{U. Stuhler}, Publ. Math., Inst. Hautes Étud. Sci. 85, 97--191 (1997; Zbl 0892.22012), and \textit{R. Bezrukavnikov}, Thesis (1998; \texttt{http://arxiv.org/abs/math/0406223})]; \textit{J. Arthur} [Acta Math. 171, 73--138 (1993; Zbl 0822.22011)] computed this pairing in terms of elliptic characters of the analytic \(R\)-group.NEWLINENEWLINEThe main result is deduced from steps including: the elliptic virtual representations of a Weyl group \(W\) are described in terms of Springer representations, using a Weyl group analogue of the \(p\)-adic pairing; the elliptic Iwahori virtual representations of \(G(F)\) are described in terms of Kazhdan-Lusztig parameters, as in [\textit{D. Kazhdan} and \textit{G. Lusztig}, Invent. Math. 87, 153--215 (1987; Zbl 0613.22004)]; Arthur's formula is shown to hold for Iwahori representations in any characteristic, using the homological definition of the pairing, and with a geometric \(R\)-group.