Spherical Hecke algebras for Kac-Moody groups over local fields

Abstract

We define the spherical Hecke algebra $\mathcal{H}$ for an almost split Kac-Moody group $G$ over a local non-archimedean field. We use the hovel $\mathscr I$ 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 $\mathcal{H}$ 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 $\mathcal{H}$ 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 $\mathcal{H}$ and the algebra of Weyl invariant elements in some completion of a Laurent polynomial algebra. In particular, $\mathcal{H}$ is always commutative. Actually, our results apply to abstract “locally finite” hovels, so that we can define the spherical algebra with unequal parameters.