Le lemme fondamental pour les algèbres de Lie

Kostant section, universal centralizer, and a modular derived Satake equivalence

“…[G1,BF,Do,Ng]. More recently, the "modular" case treated in this paper has been used in a joint work with Carl Mautner [MR2] (again, as a technical tool) to generalize some constructions of [Do] to positive characteristic, with an application to the completion of the proof of the Mirković-Vilonen conjecture on stalks of standard spherical perverse sheaves on affine Grassmannians [MV].…”

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…While the first half of §4, on toric and on semismall maps, is targeted to a general audience, the second half, on the geometrization of Hecke algebras and of the Satake isomorphism, is technically more demanding. Due to the fact that the recent and exciting development [152] in the Langlands program makes use of a result that deals with the decomposition theorem with "large fibers," we have included a brief discussion of B.C. Ngô's support theorem in §4.…”

“…of a map with large fibers. For expository reasons, we state these results over the complex numbers, even though the main use in [152] is in the l-adic context over a finite field.…”

“…The proof is a simple and elegant application of the symmetry (10) arising from Poincaré-Verdier duality, and can be found in [152], Appendice A, Théorèmes 2 and 3. The basic idea in the proof is that a bigger codimension, coupled with duality, would force the corresponding summand to contribute a nontrivial summand to the direct image sheaf R j f * Q X for j > 2d, contradicting the fact that the fibers have dimension d.…”

“…Ngô for very useful conversations. The paper [152] is devoted to the proof of the fundamental lemma in the Langland's program, a long-standing and deep conjecture concerning Lie groups. For its complexity, depth and wealth of applications to representation theory, this paper deserves a separate treatment, which we do not provide here.…”