Spherical Hecke algebras for Kac-Moody groups over local fields


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.


Stéphane Gaussent

Université de Lyon, Institut Camille Jordan (UMR 5208), Université Jean Monnet, Saint-Etienne, F-42023, France

Guy Rousseau

Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502 and CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre lès Nancy, F-54506, France