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arxiv: 2604.11735 · v1 · submitted 2026-04-13 · 🧮 math.RT · math.AG· math.NT

An analogue of irreducible cuspidal representations for the group PGL(2) over a two-dimensional local field

Alexander Braverman , David Kazhdan This is my paper

Pith reviewed 2026-05-10 15:16 UTC · model grok-4.3

classification 🧮 math.RT math.AGmath.NT
keywords PGL(2)cuspidal representationstwo-dimensional local fieldsquadratic extensionsGalois invariant charactersBorel subgroupssmooth representationsreductive groups
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The pith

Irreducible cuspidal representations of PGL(2) over two-dimensional local fields can be built from quadratic extensions and non-Galois-invariant characters.

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

This paper constructs analogues of the classical irreducible cuspidal representations of PGL(2) when the base field is a two-dimensional local field K obtained by adjoining a formal variable to a one-dimensional local field F. The construction starts with a quadratic extension L of K and a character θ from the quotient L*/K* to the complex numbers that is not preserved by the nontrivial Galois automorphism. It proves that the resulting representations restrict to irreducible representations of the Borel subgroup P(K), yet these restrictions are not the same as the standard cuspidal representations of P(K) that exist in this setting. The work also introduces a proposed definition of cuspidality that applies to smooth representations of any split reductive group over K. A sympathetic reader would care because extending such fundamental objects of representation theory to higher-dimensional local fields opens the door to studying automorphic forms and harmonic analysis in more general arithmetic contexts.

Core claim

Starting from a quadratic extension L of the two-dimensional local field K and a character θ : L^*/K^* → ℂ^* which is not Galois invariant, one obtains irreducible cuspidal representations of G(K) for G = PGL(2). The restriction of these representations to the Borel subgroup P(K) is irreducible, but unlike the classical situation over one-dimensional local fields, this restriction is not isomorphic to the standard irreducible cuspidal representation of P(K). Additionally, a general notion of cuspidality is proposed for smooth representations of H(K) where H is an arbitrary split reductive group.

What carries the argument

The map from non-Galois-invariant characters θ of L^*/K^* to representations of G(K), which serves as the analogue of the classical cuspidal construction and ensures the restriction to P(K) is irreducible yet distinct.

Load-bearing premise

The assumption that the character θ is not invariant under the Galois group action is necessary for the construction to produce a well-defined representation whose restriction to P(K) is irreducible and distinct from the standard cuspidal one, relying on the odd residue characteristic of the field.

What would settle it

Computing the restriction of the constructed representation to a specific unipotent subgroup of P(K) and checking whether it decomposes into multiple irreducible components would directly test the irreducibility claim.

read the original abstract

Let $F$ be a local non-archimedian field of odd residue characteristic and let $G=PGL(2)$. In this paper we study an analog of irreducible cuspidal representations of the group $G(F)$ when $F$ is replaced by the field $K=F((t))$. The story turns out to be similar to the classical case, but also with some differences. We present a construction of such representations essentially (up to a small subtlety) starting from a quadratic extension $L$ of $K$ and a character $\theta:L^*/K^*\to \mathbb C^*$ which is not Galois invariant. We also show that the restriction of the representations we construct to the group $P(K)$ (here $P$ is a Borel subgroup of $PGL(2)$) is irreducible. However, contrary to the classical case it turns out that these restrictions are not isomorphic to the "standard" irreducible cuspidal representation of $P(K)$. In the Appendix we propose a notion of cuspidality for smooth representations of the group $H(K)$ for an arbitrary split reductive group $H$.

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 / 2 minor

Summary. The paper constructs analogues of irreducible cuspidal representations for PGL(2) over the two-dimensional local field K=F((t)) (F a local non-archimedean field of odd residue characteristic). The construction proceeds from a quadratic extension L/K and a non-Galois-invariant character θ:L^*/K^*→ℂ^*, and the resulting representations are shown to restrict irreducibly to the Borel subgroup P(K) while being non-isomorphic to the standard cuspidal representation of P(K). An appendix proposes a general definition of cuspidality for smooth representations of H(K) for arbitrary split reductive groups H.

Significance. If the details of the construction and the irreducibility/non-isomorphism statements hold, the work provides a concrete extension of the classical cuspidal representation theory to two-dimensional local fields, explicitly identifying a difference in Borel restrictions that does not occur in the one-dimensional case. The appendix definition of cuspidality is a useful independent contribution that could support further work on representations of reductive groups over higher-dimensional local fields.

minor comments (2)
  1. [Abstract / Introduction] The abstract refers to the construction as holding 'essentially (up to a small subtlety)'; this subtlety should be stated explicitly in the introduction or §2 so that readers can assess its impact on the subsequent irreducibility argument.
  2. [Appendix] The appendix definition of cuspidality is presented independently, but a brief comparison to the classical notion for PGL(2,F) would clarify how the two-dimensional analogue aligns with or diverges from standard usage.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, the assessment of its significance, and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs analogues of cuspidal representations for PGL(2) over K=F((t)) by starting from a quadratic extension L/K and a non-Galois-invariant character θ:L^*/K^*→C^*, then proves that the resulting representations restrict irreducibly to the Borel P(K) but are non-isomorphic to the standard cuspidal representation of P(K). The appendix supplies an independent definition of cuspidality for smooth representations of H(K). All steps rely on standard properties of two-dimensional local fields of odd residue characteristic and classical representation-theoretic techniques applied to the new setting; no equation or claim reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The explicit contrast with the classical case further confirms the argument does not smuggle in its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The work rests on standard properties of local fields and Galois cohomology together with the new construction; no free parameters are fitted and the only invented item is the proposed general notion of cuspidality.

axioms (3)
  • domain assumption F is a local non-archimedean field of odd residue characteristic
    Used to define the base field and the two-dimensional extension K = F((t))
  • standard math Existence and basic properties of quadratic extensions L of K
    Invoked to start the construction from L and the quotient L^*/K^*
  • domain assumption Characters θ : L^*/K^* → C^* that are not Galois invariant exist and behave as in the classical case
    Central hypothesis for the construction to produce the desired representations
invented entities (1)
  • General notion of cuspidality for smooth representations of H(K) no independent evidence
    purpose: To define cuspidality uniformly for arbitrary split reductive groups over two-dimensional local fields
    Proposed in the appendix as a new definition extending the classical notion

pith-pipeline@v0.9.0 · 5514 in / 1663 out tokens · 90217 ms · 2026-05-10T15:16:06.553377+00:00 · methodology

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