Spherical Hecke algebras for Kac-Moody groups over local fields

Details Spherical Hecke algebras for Kac-Moody groups over local fields Spherical Hecke algebras for Kac-Moody groups over local fields

2012-01-29

arxiv.org/abs/1201.6050v1

Mathematics

Rings and Algebras

Scientific paper

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. Actually, our results apply to abstract "locally finite" hovels. When G is split, we prove that H is commutative.

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