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arxiv: 2510.24201 · v2 · submitted 2025-10-28 · 🧮 math.RT

Rationality properties of complex representations of reductive p-adic groups

David Kazhdan , Maarten Solleveld , Yakov Varshavsky This is my paper

Pith reviewed 2026-05-18 03:48 UTC · model grok-4.3

classification 🧮 math.RT
keywords elliptic representationsp-adic reductive groupsrationalitycentral charactersquare-integrable representationsGalois actionsmooth representations
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The pith

An elliptic representation of a reductive p-adic group realizes over Qbar if and only if its central character takes values in Qbar.

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

The paper compares smooth representations of reductive groups over non-archimedean local fields when the coefficients are the complex numbers versus an algebraic closure of the rationals. It establishes that an elliptic representation in Arthur's sense admits a model over Qbar precisely when the central character is defined over Qbar, and this criterion covers all essentially square-integrable representations as a special case. The result also shows that both the elliptic representations and the essentially square-integrable ones remain unchanged as sets when the Galois group of C over Q acts on them.

Core claim

For a reductive group G over a non-archimedean local field, an elliptic G-representation in the sense of Arthur can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. The automorphism group of C over Q preserves the sets of essentially square-integrable representations and of elliptic representations.

What carries the argument

The rationality criterion for elliptic representations that links realizability over Qbar directly to the values of the central character, together with the Galois stability of the relevant sets.

If this is right

  • Every essentially square-integrable representation satisfies the same realizability criterion over Qbar.
  • The full collection of elliptic representations is invariant under the action of Gal(C/Q).
  • The full collection of essentially square-integrable representations is likewise invariant under Gal(C/Q).
  • Any representation whose central character lies in Qbar admits a realization with coefficients in Qbar.

Where Pith is reading between the lines

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

  • The criterion may allow descent of character computations or matrix coefficients to number fields for many representations.
  • For concrete groups such as GL(n), explicit lists of elliptic representations could be checked directly against the central-character condition.
  • The Galois stability suggests that arithmetic invariants of these representations are already visible over finite extensions of Q.

Load-bearing premise

The standard definitions and expected properties of elliptic representations in Arthur's sense and of smooth representations of reductive p-adic groups hold without exception.

What would settle it

An explicit elliptic representation whose central character does not take values in Qbar but which nevertheless admits a model over Qbar, or an elliptic representation with central character in Qbar that has no model over Qbar.

read the original abstract

For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). We show that an elliptic G-representation (in the sense of Arthur) can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. We also study the action of the automorphism group of C/Q on complex G-representations. We prove that the sets of essentially square-integrable representations and of elliptic representations are stable under Gal(C/Q).

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.

Referee Report

0 major / 3 minor

Summary. The paper compares smooth representations of reductive groups G over non-archimedean local fields with coefficients in C versus Qbar. It proves that an elliptic G-representation in Arthur's sense can be realized over Qbar if and only if its central character takes values in Qbar; this applies in particular to all essentially square-integrable representations. It further shows that the sets of essentially square-integrable representations and of elliptic representations are stable under the natural action of Gal(C/Q).

Significance. If the results hold, they establish a precise rationality criterion for elliptic and square-integrable representations via the Bernstein center (defined over Q) and Galois stability of the elliptic support condition on character distributions. This provides a useful descent tool for representations arising in the local Langlands correspondence and automorphic forms, and the stability statements clarify the Galois-equivariance of these classes.

minor comments (3)
  1. [§2] §2: the statement of the main theorem would be clearer if the precise definition of 'realized over Qbar' (i.e., existence of a G-stable Qbar-lattice in the underlying vector space) were recalled explicitly before the proof.
  2. The notation for the Bernstein center and its action on elliptic representations is introduced without a forward reference to the relevant theorem in the literature; adding one or two citations would improve readability.
  3. [§4] The proof of Galois stability for elliptic representations relies on the character distribution being defined over Q; a short remark on how this interacts with non-split tori would help readers working with general reductive groups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of our results, and recommendation of minor revision. We have no major comments to address point-by-point as none were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central theorem states that an elliptic representation (in Arthur's sense) is realizable over Qbar precisely when its central character takes values in Qbar, with the only-if direction immediate from the definition of a Qbar-model and the if direction following from the action of the Bernstein center (defined over Q) together with descent of the module. The stability of essentially square-integrable and elliptic representations under Gal(C/Q) is shown by verifying that the elliptic support condition on the character distribution is preserved under coefficient-wise Galois action. These steps rely on the standard setup of smooth representations of reductive groups over non-archimedean local fields and established properties of the Bernstein center, without any reduction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations whose content is unverified. The argument is therefore independent of the target result and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard axioms of the representation theory of reductive p-adic groups (smooth representations, Bernstein center, types) and on Arthur's definition of elliptic representations; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Standard properties of smooth representations of reductive groups over non-archimedean local fields and the Bernstein center hold as in the literature.
    Invoked implicitly to compare representations over C and over Qbar.
  • domain assumption Arthur's definition of elliptic representations is the one used throughout.
    Central to the statement of the main theorem.

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discussion (0)

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