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Perverse sheaves, base change homomorphism, and fundamental lemma of Jacquet and Ye. (Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye.) (French) Zbl 1002.11046

Summary: We give a geometric interpretation of the base change homomorphism between the Hecke algebra of \(\text{GL}(n)\) for an unramified extension of local fields of positive characteristic. For this, we use some results of V. Ginzburg [Perverse sheaves on a loop group and Langlands duality. Preprint alg-geom, 9511007 (1995)], I. Mirkovic and K. Vilonen [Perverse sheaves on loop Grassmannians and Langlands duality. Preprint alg-geom, 9703010 (1997), see also Math. Res. Lett. 7, 13–24 (2000; Zbl 0987.14015)] related to the geometric Satake isomorphism. We give a new proof of these results in the positive characteristic case.
By using that geometric interpretation of the base change homomorphism, we prove the fundamental lemma of H. Jacquet and Y. Ye [C. R. Acad. Sci., Paris, Sér. I 311, 671–676 (1990; Zbl 0715.11026)] for an arbitrary Hecke function in the case of equal characteristic.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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References:

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