Iwahori-Hecke algebras for Kac-Moody groups over local fields

TL;DR

This paper introduces a new Iwahori-Hecke algebra for almost split Kac-Moody groups over local fields, extending the classical theory to more general groups using hovels, and relates it to known algebraic structures like Cherednik's double affine Hecke algebra.

Contribution

It defines the Iwahori-Hecke algebra for Kac-Moody groups over local fields, provides a presentation similar to the Bernstein-Lusztig form, and connects it to existing algebraic frameworks.

Findings

Structure constants are polynomials in the residue field size.

The algebra embeds into a larger algebra with Bernstein-Lusztig presentation.

In the affine case, it contains Cherednik's double affine Hecke algebra.

Abstract

We define the Iwahori-Hecke algebra for an almost split Kac-Moody group over a local non-archimedean field. We use the hovel associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The fixer K of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define the Iwahori-Hecke algebra as the algebra of some K-bi-invariant functions on the group with support consisting of a finite union of double classes. As two chambers in the hovel are not always in a same apartment, this support has to be in some large subsemigroup of the Kac-Moody group. In the split case, we prove that the structure constants of the multiplication in this algebra are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We give a presentation of this algebra, similar to the Bernstein-Lusztig presentation in the reductive case, and embed it in a greater algebra, algebraically defined by the Bernstein-Lusztig presentation. In the affine case, this algebra contains the Cherednik's double affine Hecke algebra. Actually, our results apply to abstract "locally finite" hovels, so that we can define the Iwahori-Hecke algebra with unequal parameters.