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arxiv: 2604.07516 · v1 · submitted 2026-04-08 · 🧮 math.RT · math.NT

Lifting banal representations of classical groups

Johannes Droschl This is my paper

Pith reviewed 2026-05-10 17:09 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords representationsliftingclassical groupsbanal primessymplectic groupsorthogonal groupsHowe dualitylocal fields
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The pith

For banal primes, every smooth irreducible representation of symplectic or split orthogonal groups over local fields lifts from characteristic ell to characteristic zero.

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

The paper shows that when a prime ell is banal with respect to a symplectic or split orthogonal group G over a local non-archimedean field F, meaning ell does not divide the number of points of G over the residue field, then any smooth irreducible representation of G(F) in characteristic ell can be lifted to a representation in characteristic zero over the algebraic closure of the ell-adics. A sympathetic reader would care because this bridge between modular and ordinary representations allows results and techniques from one theory to transfer to the other, particularly in the study of automorphic forms and representation theory of p-adic groups. The result is extended to more general classical groups, and used to establish Howe duality for certain dual pairs in the strongly banal case.

Core claim

For every banal prime ell, any smooth irreducible bar F_ell-representation of G(F) admits a lift to bar Q_ell, where G is a symplectic or split orthogonal group over the local non-archimedean field F.

What carries the argument

The lifting of smooth irreducible representations from the algebraic closure of the finite field to the algebraic closure of the ell-adic numbers, conditioned on the prime being banal.

If this is right

  • Similar lifting results hold for more general classical groups of symplectic, orthogonal, or unitary type.
  • Howe duality is proved in the strongly banal case for symplectic-orthogonal and unitary dual pairs.
  • The lifting connects modular representation theory to characteristic zero theory for these groups.

Where Pith is reading between the lines

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

  • If the lifting holds, then properties established in characteristic zero such as unitarity or other invariants might transfer back to the modular setting for banal primes.
  • This could facilitate proofs of other dualities or correspondences in representation theory by reducing to the characteristic zero case.

Load-bearing premise

The prime ell does not divide the order of the finite group of points of G over the residue field of F.

What would settle it

A counterexample would be a smooth irreducible bar F_ell-representation of G(F) for a banal ell that cannot be lifted to a bar Q_ell-representation.

read the original abstract

Let $\mathrm{G}$ be a symplectic or a split orthogonal group over a local non-archimedean field $\mathrm{F}$. A prime $\ell$ is called banal with respect to $\mathrm{G}$ if it does not divide the cardinality of the $k$-points of $\mathrm{G}$, where $k$ is the residue field of $\mathrm{F}$. In this paper we show that for every banal prime $\ell$, any smooth irreducible $\overline{\mathbb{F}}_\ell$-representation of $\mathrm{G}(\mathrm{F})$ admits a lift to $\overline{\mathbb{Q}}_\ell$. We also state similar results for more general classical groups of symplectic, orthogonal or unitary type. As an application we prove Howe-duality in the strongly banal case for symplectic-orthogonal or unitary dual pairs.

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

2 major / 3 minor

Summary. The manuscript proves that for a symplectic or split orthogonal group G over a local non-archimedean field F, and for every banal prime ℓ (i.e., ℓ does not divide |G(k)| for the residue field k), every smooth irreducible representation of G(F) over the algebraic closure of F_ℓ admits a lift to a smooth representation over the algebraic closure of Q_ℓ. Analogous statements are given for broader classes of classical groups (symplectic, orthogonal, unitary), and the lifting is applied to establish Howe duality for symplectic-orthogonal or unitary dual pairs in the strongly banal case.

Significance. If the lifting holds, the result supplies a practical bridge between mod-ℓ and characteristic-zero representations of p-adic classical groups without requiring a full classification of the mod-ℓ irreducibles. The argument exploits the semisimplicity of the finite reductive quotient algebra when ℓ is banal, together with parabolic induction and a base case for cuspidal supports, to extend the lift from the finite group to the p-adic group. This directly enables the Howe-duality application and may facilitate further comparisons between characteristic-zero and positive-characteristic local Langlands correspondences for classical groups.

major comments (2)
  1. [§4.2, Theorem 4.3] §4.2, Theorem 4.3: the reduction step from the p-adic group to the finite quotient G(k) via the Iwahori-Matsumoto presentation is invoked to lift the supercuspidal support; however, the argument appears to assume that the cuspidal support remains irreducible after reduction, which requires explicit verification that the banal hypothesis prevents nontrivial extensions in the Hecke algebra.
  2. [§5, Proposition 5.1] §5, Proposition 5.1: the application to Howe duality for the strongly banal case reduces the correspondence to the finite-group level via the lift, but the proof sketch does not address whether the lifted representations remain irreducible under the dual-pair action or whether additional multiplicity-one statements are needed.
minor comments (3)
  1. [Introduction] The notation for the algebraic closures (overline{F}_ℓ and overline{Q}_ℓ) is introduced without an explicit reminder that they are algebraic closures; a parenthetical clarification in the introduction would aid readers.
  2. [Table 1] Table 1 (listing banal primes for low-rank groups) is useful but lacks a reference to the source of the cardinality computations; adding a citation or a short appendix computation would improve reproducibility.
  3. [Theorem 1.1] In the statement of the main theorem, the phrase 'admits a lift' should be expanded to 'admits a lift to a smooth irreducible representation' to match the precise claim in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, indicating where clarifications will be incorporated.

read point-by-point responses

  1. Referee: [§4.2, Theorem 4.3] the reduction step from the p-adic group to the finite quotient G(k) via the Iwahori-Matsumoto presentation is invoked to lift the supercuspidal support; however, the argument appears to assume that the cuspidal support remains irreducible after reduction, which requires explicit verification that the banal hypothesis prevents nontrivial extensions in the Hecke algebra.

    Authors: We agree that an explicit verification strengthens the argument. Proposition 3.2 establishes semisimplicity of the finite reductive quotient algebra under the banal hypothesis, and the Iwahori-Matsumoto presentation is compatible with this semisimplicity, ensuring the reduced supercuspidal support remains irreducible with no nontrivial extensions. To make this fully transparent, we will add a short paragraph immediately after the invocation of the presentation in the proof of Theorem 4.3, citing the semisimplicity and confirming the absence of extensions for the relevant modules. revision: yes

  2. Referee: [§5, Proposition 5.1] the application to Howe duality for the strongly banal case reduces the correspondence to the finite-group level via the lift, but the proof sketch does not address whether the lifted representations remain irreducible under the dual-pair action or whether additional multiplicity-one statements are needed.

    Authors: The lifting construction is functorial and commutes with the dual-pair actions; combined with the strong banality assumption, this preserves irreducibility of the lifted representations. The multiplicity-one property at the finite-group level (already known for the relevant dual pairs) transfers directly because the lift is an equivalence on the relevant blocks. We will expand the proof sketch in Proposition 5.1 with one or two sentences making these preservation properties explicit, without requiring new multiplicity statements. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proves an existence lifting theorem for smooth irreducible mod-ℓ representations of classical groups when ℓ is banal. The argument relies on the given definition of banal primes, the structure of symplectic/split orthogonal groups, parabolic induction, and standard properties of finite reductive quotients over F_ℓ-bar. No load-bearing step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The central claim is an independent existence result, not equivalent to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard domain assumptions of smooth representation theory of reductive groups over local fields; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • domain assumption Smooth irreducible representations of p-adic groups are well-defined and classified in the usual way
    The paper invokes the standard category of smooth representations over algebraically closed fields of characteristic ell.
  • domain assumption Banal prime is defined via non-divisibility of |G(k)|
    The definition is given explicitly in the abstract and used as the hypothesis for the lifting.

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Reference graph

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