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On endoscopic transfer of Deligne-Lusztig functions. (English) Zbl 1241.22021

In the representation theory of \(p\)-adic groups, the notions of endoscopic groups and endoscopic transfer are important. Let \(G\) be a connected, reductive algebraic group over a \(p\)-adic field \(F\), and let \(H\) be an endoscopic subgroup (for \(SL(2)\) for instance, there are three such subgroups: \(SL(2)\) itself, \(\mathbf{G}_m\) and \(U_E\), a 1-dimensional torus splitting over a quadratic extension \(E\) of \(F\)). Results due to Langlands, Shelstad, Ngô and Waldspurger imply that corresponding to any \(\phi \in C^{\infty}_c(G(F))\), there exists \(\phi^H \in C^{\infty}_c(H(F))\) which has “matching orbital integrals”. Such a \(\phi^H\) is called an endoscopic transfer of \(\phi\). Although an endoscopic transfer is not unique in general, Langlands conjectured (known since then as “the fundamental lemma”) that for spherical functions \(\phi\), there is a unique spherical endoscopic transfer. The fundamental lemma was proved by a combination of results due to Hales, Ngô and Waldspurger; it is even true that \(\phi \mapsto \phi^H\) is an explicit algebra homomorphism.
It is expected even in general that for all “interesting” functions \(\phi\), one can define explicitly the “best” endoscopic transfer. In 1997, Kottwitz suggested a conjectural candidate for \(\phi^H\) when \(G\) is split, adjoint (in this case \(H\) is necessarily split) and \(\phi\) is supported on \(G(O_F)\) and equals the inflation of the character of a unipotent Deligne-Lusztig virtual representation of \(G(\mathbf{f})\), where \(\mathbf{f}\) is the residue field of \(F\). He conjectured that \(\phi^H\) is an explicit linear combination of unipotent Deligne-Lusztig functions for \(H(F)\), supported on \(H(O_F)\).
This important paper proves a generalization of Kottwitz’s conjecture under the (mild) hypothesis that the residue characteristic does not divide the order of the Weyl group of \(G\). The authors use a theorem of Waldspurger whose proof is global, apart from which the arguments of the paper are local. The main theorem of the paper also implies the fundamental lemma for the characteristic function of the Iwahori subgroup as well as the fundamental lemma for unit elements. However, as the authors point out, it is not a new proof of the fundamental lemma for unit elements since they use Waldspurger’s theorem whose proof already used the fundamental lemma for unit elements. There are also some results of independent interest; for instance, the authors prove (a result apparently known to specialists) that the local Langlands correspondence for tori is compatible with field extensions.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
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