Spherical Hecke algebras for Kac-Moody groups over local fields

• Autores: Stéphane Gaussent
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 180, Nº 3, 2014, págs. 1051-1087
• Idioma: inglés
• DOI: 10.4007/annals.2014.180.3.5
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• Resumen

  • We define the spherical Hecke algebra H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The stabilizer K of a special point on the standard apartment plays the role of a maximal open compact subgroup. We can define H as the algebra of K -bi-invariant functions on G with almost finite support. As two points in the hovel are not always in a same apartment, this support has to be in some large subsemigroup G + of G . We prove that the structure constants of H are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We also prove the Satake isomorphism between H and the algebra of Weyl invariant elements in some completion of a Laurent polynomial algebra. In particular, H is always commutative. Actually, our results apply to abstract �locally finite� hovels, so that we can define the spherical algebra with unequal parameters