Spherical Hecke algebras for Kac-Moody groups over local fields

Stéphane Gaussent (1) , Guy Rousseau (1)

1 IECN - Institut Élie Cartan de Nancy

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.

Mots clés: Algèbre de Hecke sphérique; Groupes de Kac-Moody; Masures

Domaines: Anneaux et algèbres [math.RA]; Théorie des représentations [math.RT]

Stéphane Gaussent, Guy Rousseau. Spherical Hecke algebras for Kac-Moody groups over local fields. 2012. ⟨hal-00664141v1⟩