## 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

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)].

### MSC:

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

### Citations:

Zbl 0666.22003; Zbl 0532.22017; Zbl 0651.22010; Zbl 0538.20014
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