Sur les intégrales orbitales tordues pour les groupes linéaires: Un lemme fondamental. (On twisted orbital integrals for linear groups: A fundamental lemma). (French) Zbl 0760.22026

Waldspurger, J.-L.

Can. J. Math. 43, No. 4, 852-896 (1991).

The proper context for understanding what this paper accomplishes is the so-called fundamental lemma conjectured in the theory of endoscopy. We recall that to each pair of endoscopic groups \((G,H)\) over a local field \(F\) there is a transfer problem for orbital integrals: to a function \(f\) on \(G\) we must associate a function \(f^ H\) on \(H\) with matching orbital integrals, i.e., \[ \Delta^ H_ G(\gamma)\Phi^ \kappa(\gamma,f)=\Phi^{st}(\gamma_ H,f^ H) \] for pairs \((\gamma,\gamma_ H)\) of associated regular semi-simple elements in \(G\) and \(H\). (Here \(\Phi^ \kappa(\gamma,f)\) and \(\Phi^{st}(\gamma_ H,f^ H)\) are the “kappa” and “stable” orbital integrals, respectively, and \(\Delta^ H_ G(\gamma)\) is a “transfer factor” [cf. R. P. Langlands “Les débuts d’une formula des traces stable” (Publ. Math. Univ. Paris VII, 13, 1982; Zbl 0532.22017)].) On the other hand, if \(F\) is \(p\)-adic, and the \(L\)-map \(\varphi:{^ LH}\to{^ LG}\) is “unramified”, then \(\varphi\) gives rise to a homomorphism \(b:{\mathcal H}(G)\to{\mathcal H}(H)\) between the Hecke algebras of \(G\) and \(H\), and the fundamental lemma asserts that (for \(f\) in \({\mathcal H}(G))\) \(b(f)\) matches with \(f\), i.e., equals \(f^ H\).
The present paper deduces this lemma in the context of \(H=GL(m,E)\), where \(E\) is an extension of \(F\) of degree \(r\), and \(G=GL(mr,F)\). (More precisely, as the author points out, this pair \((H,G)\) is not quite an endoscopic pair, but \(H^*=\{(h,x)\in H\times E^*:N_{E/F}(\text{det}(h)x)=1\}\), \(G^*=\{(g,x)\in G\times E^*:(\text{det }g)N_{E/F}(x)=1\}\) is.) Not unexpectedly, the proof is a tour de force which involves establishing recurrence formulas for the integrals \(\Phi^ \kappa(\gamma,f)\) by way of their (Shalika) germ expansions, and generalizes the “non-twisted” \((\kappa=1)\) case treated earlier by the author [Math. Ann. 284, 199-221 (1989; Zbl 0651.22010)]. The resulting fundamental lemma alluded to above generalizes the already significant case \(m=1\) treated earlier by D. Kazhdan [Lect. Notes Math. 1041, 209-249 (1984; Zbl 0538.20014)].

Reviewer: S.Gelbart (Rehovot)

MSC:

22E50	Representations of Lie and linear algebraic groups over local fields
11F85	\(p\)-adic theory, local fields

Keywords:

transfer factor; fundamental lemma; endoscopic groups; orbital integrals; Hecke algebras; recurrence formulas; (Shalika) germ expansions

Citations:

Zbl 0666.22003; Zbl 0532.22017; Zbl 0651.22010; Zbl 0538.20014