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

Nicole Bardy-Panse, Stéphane Gaussent and Guy Rousseau

Vol. 285 (2016), No. 1, 1–61

DOI: 10.2140/pjm.2016.285.1

Abstract

We define the Iwahori–Hecke algebra ${}^I\mathcal{\mathscr{H}}$ for an almost split Kac–Moody group $G$ over a local nonarchimedean field. We use the hovel $\mathcal{ℐ}$ associated to this situation, which is the analogue of the Bruhat–Tits building for a reductive group. The fixer $K_I$ of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define ${}^I\mathcal{\mathscr{H}}$ as the algebra of some $K_I$-bi-invariant functions on $G$ 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 $G^{+}$ of $G$. In the split case, we prove that the structure constants of ${}^I\mathcal{\mathscr{H}}$ 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 ${}^{BL}\mathcal{\mathscr{H}}$, algebraically defined by the Bernstein–Lusztig presentation. In the affine case, this algebra ${}^{BL}\mathcal{\mathscr{H}}$ 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.

Keywords

hovels, Hecke algebras, Bernstein–Lusztig relations, Kac–Moody groups, local fields

Mathematical Subject Classification 2010

Primary: 17B67, 20C08, 20G44

Secondary: 22E65, 33D80, 51E24, 20E42, 22E50

Milestones

Received: 4 June 2015

Revised: 17 March 2016

Accepted: 9 May 2016

Published: 27 September 2016