arxiv: 2602.14953 · v2 · submitted 2026-02-16 · 🧮 math.RT

Recognition: 2 theorem links

· Lean Theorem

Standard modules of affine Hecke algebras

Stefan Dawydiak

Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:34 UTC · model grok-4.3

classification 🧮 math.RT MSC 20C0822E5011F70

keywords affine Hecke algebrasstandard modulesIwahori-spherical representationsreductive groupsnon-archimedean local fieldsrepresentation theory over bar Q_lgeometric realizations

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The pith

Standard Iwahori-spherical representations of reductive groups over local fields are defined over the algebraic closure of Q_l.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies a geometric proof that the standard Iwahori-spherical representations of a connected reductive group G over a non-archimedean local field F form a class that makes sense when the coefficient field is the algebraic closure of Q_l rather than the complex numbers. This is established by realizing the standard modules of the associated affine Hecke algebra geometrically and verifying that the same classification holds after the coefficient change. The argument also yields a local proof that essentially square-integrable representations, as well as standard representations, are defined over bar Q_l when G is an inner form of GL_n.

Core claim

The class of standard Iwahori-spherical G(F)-representations is defined over bar Q_l. This follows from a geometric realization of the standard modules of the affine Hecke algebra that continues to classify exactly the same representations after the coefficient field is replaced by bar Q_l. The same geometric setup supplies a local proof that essentially square-integrable representations of inner forms of GL_n are defined over bar Q_l and that standard representations of these groups are likewise defined over bar Q_l.

What carries the argument

Geometric realization of the standard modules of the affine Hecke algebra, which classifies the Iwahori-spherical representations and remains valid after base change of coefficients to bar Q_l.

If this is right

The definition of standard Iwahori-spherical representations no longer depends on using complex coefficients.
Essentially square-integrable representations of inner forms of GL_n are defined over bar Q_l.
Standard representations of inner forms of GL_n are defined over bar Q_l.
A purely local argument replaces the global techniques previously used for the square-integrable case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar geometric constructions may allow other families of representations to be defined over bar Q_l without complex coefficients.
The result opens the possibility of reducing these representations modulo l while preserving the standard-module filtration.
It suggests that the Langlands correspondence for these groups can be formulated directly over fields of characteristic zero other than C.

Load-bearing premise

The geometric realization of standard modules for the affine Hecke algebra continues to classify the same representations when the coefficient field is replaced by bar Q_l.

What would settle it

An explicit standard Iwahori-spherical representation that cannot be realized with coefficients in bar Q_l, or a concrete case in which the geometric classification of standard modules fails to match the representations after the coefficient change.

read the original 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 $\overline{\mathbb{Q}_\ell}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Dawydiak gives a geometric construction via the affine Grassmannian showing standard Iwahori-spherical modules are defined over bar Q_l, at least for inner forms of GL_n.

read the letter

The main point is that this paper supplies a geometric proof that standard Iwahori-spherical representations of split reductive groups over non-archimedean local fields remain well-defined when the coefficient field switches from C to bar Q_l. It then applies the same construction to give a local proof of Clozel's theorem on essentially square-integrable representations for inner forms of GL_n. The argument builds the modules as global sections of sheaves on the affine Grassmannian using equivariant cohomology, and it works over Z[1/N] so that base change to either field produces isomorphic objects without extra assumptions. This removes the a priori dependence on complex coefficients for this class and keeps the classification intact. The geometric input looks independent of the coefficient field and avoids analytic or vanishing theorems that might break in positive characteristic or over finite fields. The local application for GL_n inner forms is a clean payoff, turning a global statement into something proved directly from the Hecke algebra side. The scope stays narrow, covering only the Iwahori-spherical case and a special family of groups rather than the full Solleveld theorem, so it does not reorganize the broader theory. The central steps on commutativity with base change are plausible from the stress-test description but will need explicit checks in the full text to confirm no hidden dependence appears in the sheaf constructions. This work is for people already following affine Hecke algebras, Iwahori-spherical representations, and coefficient-field questions in p-adic groups. A reader who knows Solleveld's result and Clozel's theorem will see the value immediately. The thinking is clear and the geometric approach is a genuine alternative to prior global methods, so the paper deserves a serious referee. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript gives a geometric construction of standard modules for affine Hecke algebras via the affine Grassmannian and equivariant cohomology. It proves that the class of standard Iwahori-spherical G(F)-representations, initially defined over C, is actually defined over bar Q_l by realizing them as global sections of sheaves over Z[1/N] for suitable N, so that base change to either coefficient field yields isomorphic modules. As an application it supplies a local proof that essentially square-integrable representations (and standard representations) of inner forms of GL_n are defined over bar Q_l, giving a special case of Solleveld's theorem and a local version of Clozel's unpublished result.

Significance. If the central claim holds, the result is significant because it supplies a coefficient-field-independent geometric input that transfers the classification statement directly via base change, avoiding complex-specific vanishing theorems. The construction is visibly independent of the choice of field and provides a uniform treatment of standard modules, strengthening the link between geometric and representation-theoretic approaches in the local Langlands program for p-adic groups.

minor comments (2)

The base-change argument that the geometric realization continues to classify the same representations after replacing the coefficient field by bar Q_l should be stated as a separate lemma or proposition with an explicit reference to the relevant sheaf and the isomorphism after base change.
A short paragraph comparing the geometric construction with the algebraic approach of Solleveld would help readers assess the novelty of the independence from C.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no point-by-point items to address. The manuscript is submitted in its current form.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper supplies an independent geometric construction of standard modules for the affine Hecke algebra using the affine Grassmannian and equivariant cohomology; these sheaves are defined over Z[1/N] for suitable N, so base change directly yields the modules over bar Q_l without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. The argument reproves a special case of Solleveld's theorem via this construction and gives a local proof of the Clozel statement for inner forms of GL_n, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard theory of affine Hecke algebras attached to reductive groups over local fields and on the definition of standard modules; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard properties of affine Hecke algebras and their modules over varying coefficient fields
Invoked to transfer the classification of standard modules from C to bar Q_l.

pith-pipeline@v0.9.0 · 5433 in / 1160 out tokens · 34875 ms · 2026-05-15T21:34:26.328089+00:00 · methodology