Standard modules of affine Hecke algebras

TL;DR

The paper develops a geometric framework showing that standard modules for $G(F)$, central to the Local Langlands program, are defined over $\bar{\mathbb{Q}}_\ell$ and behave well under field automorphisms. By realizing standard modules in Chow groups and employing a compatibility with the Bernstein subalgebra, the author proves a precise twist relation $\gamma\cdot K(s,N,\rho) \cong K(\gamma(s),N,\rho_\gamma)$, and shows that essential square-integrability and formal degrees are preserved under $\gamma$-twists. In the GL$_n$ case, a local proof of Clozel’s unpublished theorem is given by translating representations to affine Hecke-algebra modules and leveraging commutative diagrams and known results on degrees. The work connects geometric representation theory with categorical Langlands expectations, offering a robust approach that extends to principal blocks and potentially unipotent blocks in future work. Overall, the paper strengthens the understanding of how standard and discrete series representations behave under automorphisms and provides concrete tools for analyzing base-change phenomena in $p$-adic representation theory.

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 $\bar{\mathbb{Q}}_\ell$. An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially square-integrable representations is also defined over $\bar{\mathbb{Q}}_\ell$. As an application of our main result, we give a local proof of this theorem for $G=\mathrm{GL}_n$.

Theorems & Definitions (22)

• Theorem 1: SolleveldStandard, Prop. 5.5
• Theorem 2: Clozel, unpublished
• Theorem 3: DLP
• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof