Standard modules of affine Hecke algebras

TL;DR

The paper provides a geometric proof that certain representations of split reductive groups over non-archimedean fields are defined over algebraic closures of , with applications to inner forms of .

Contribution

It introduces a new geometric proof establishing that standard Iwahori-spherical representations are defined over , extending to inner forms of .

Findings

Standard Iwahori-spherical representations are defined over .

Unpublished Clozel theorem on square-integrable representations is reproved locally.

Standard representations of inner forms of  are also shown to be defined over .

Abstract

Let $G$ be a connected reductive group defined and split over a non-archimedean local field $F$. We give a new geometric proof of a special case of a recent theorem of Solleveld. Namely, we show that the class of standard Iwahori-spherical $G(F)$-representations, a notion a priori dependent on the coefficient field being the complex numbers, is actually defined over $\overline{\mathbb{Q}_\ell}$. An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially square-integrable representations is also defined over $\overline{\mathbb{Q}_\ell}$. As an application of our main result, we give a local proof of this theorem for inner forms of $\mathrm{GL}_n$, as well as showing that standard representations of these groups are defined over $\overline{\mathbb{Q}_\ell}$.