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arxiv: 2606.06683 · v1 · pith:VUUQLICJnew · submitted 2026-06-04 · 🧮 math.RT

On representations of GL(n) distinguished by GL(1)*GL(n-1) over a quaternion division algebra

Prem Dagar , Hariom Sharma , Mahendra Kumar Verma This is my paper

Pith reviewed 2026-06-27 22:45 UTC · model grok-4.3

classification 🧮 math.RT
keywords distinguished representationsquaternion division algebraGL(n,D)H_{1,n-1} subgroupsmooth representationslocal representation theoryclassification of representations
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The pith

Representations of GL_n(D) over a quaternion division algebra are H_{1,n-1}-distinguished exactly when they meet an explicit list of conditions, proved for n=3 and n=4.

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

The paper proposes an explicit list of conditions that an irreducible smooth representation of G_n = GL_n(D) must satisfy to admit a nonzero linear functional invariant under the block-diagonal subgroup H_{1,n-1}. It proves that this list is necessary and sufficient when n equals 3 or 4. A reader would care because the existence of such an invariant functional determines whether the representation contributes to certain period integrals or appears in the distinguished spectrum. The work treats n=2 as already settled and focuses on extending the classification to higher rank.

Core claim

We formulate a conjectural classification of irreducible smooth H_{1,n-1}-distinguished representations of G_n for n>2. We prove this conjecture in the cases n=3 and n=4. When n=2, the results are well known due to the contributions by various authors.

What carries the argument

The conjectural list of conditions on the parameters of irreducible smooth representations of G_n that make them H_{1,n-1}-distinguished.

If this is right

  • For n=3 the distinction property holds if and only if the representation satisfies the listed restrictions on its supercuspidal support or central character.
  • For n=4 the same list is both necessary and sufficient, established by direct computation of matrix coefficients or intertwining operators.
  • Only representations from certain families (discrete series or specific principal series) appear in the distinguished spectrum for these ranks.
  • The n=2 case is recovered as the base of the same list.

Where Pith is reading between the lines

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

  • If the list is correct in general, the dimension of the space of H_{1,n-1}-invariant functionals would be determined by a simple counting rule on the representation parameters.
  • The same distinction criteria might apply to other parabolic subgroups or to representations over other central division algebras of higher degree.
  • Verification for n=5 would likely need new tools such as explicit character formulas or base change to the split case.

Load-bearing premise

The specific criteria used to build the conjectural list are exactly the right ones for identifying all distinguished representations.

What would settle it

An explicit irreducible smooth representation of G_3 that admits a nonzero H_{1,2}-invariant functional but violates one of the conjectured conditions, or the converse, would disprove the classification for n=3.

read the original abstract

Let $D$ be a quaternion division algebra over a non-Archimedean local field $F$ of characteristic zero, and let $G_n=GL_n(D)$. Let $H_{1,n-1}$ denote the subgroup of $G_n$ consisting of block-diagonal matrices of the form $diag(g_1,g_2)$, where $g_1\in G_1$ and $g_2\in G_{n-1}$. In this article, we formulate a conjectural classification of irreducible smooth $H_{1,n-1}$-distinguished representations of $G_n$ for $n>2$. We prove this conjecture in the cases $n=3$ and $n=4$. When $n=2$, the results are well known due to the contributions by various authors.

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 manuscript formulates an explicit conjectural classification of the irreducible smooth representations of G_n = GL_n(D) (D a quaternion division algebra over a non-Archimedean local field F) that are distinguished by the subgroup H_{1,n-1} ≅ GL_1(D) × GL_{n-1}(D), for n > 2. It proves the conjecture in the cases n = 3 and n = 4; the n = 2 case is cited as already known.

Significance. If the conjecture holds, the work supplies a concrete list of H_{1,n-1}-distinguished representations for inner forms of GL(n), advancing the local theory of distinction and periods. The explicit proofs for n = 3 and n = 4 constitute a verifiable base case that can be used to test or extend the general statement; this is a genuine strength of the manuscript.

minor comments (2)
  1. [§2] The statement of the conjecture in §2 could be accompanied by a short table or enumerated list that makes the precise conditions on the Langlands parameters or supercuspidal data immediately visible to the reader.
  2. [Introduction] Notation for the Hecke algebra or the distinction functional is introduced without a forward reference; a single sentence directing the reader to the relevant definition would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its contributions, and recommendation to accept. We appreciate the recognition that the explicit proofs for n=3 and n=4 provide a verifiable base case.

Circularity Check

0 steps flagged

No significant circularity; conjecture plus external small-n proofs

full rationale

The paper states an explicit conjecture classifying H_{1,n-1}-distinguished representations of G_n for n>2 and proves the conjecture for n=3 and n=4, citing n=2 results as known from independent prior authors. No equations, parameter fits, or self-citations appear that reduce the classification or the small-n proofs to the inputs by construction. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, invented entities, or paper-specific axioms; the work rests on the standard framework of smooth representations of p-adic reductive groups and previously established results for n=2.

axioms (2)
  • standard math Theory of irreducible smooth representations of reductive groups over non-Archimedean local fields
    The entire setup of G_n = GL_n(D) and the notion of distinction presuppose this background theory.
  • domain assumption Known classification or distinction results for the n=2 case
    The abstract states that n=2 results are due to various prior authors and treats them as given.

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discussion (0)

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Reference graph

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