arxiv: 2605.03519 · v1 · submitted 2026-05-05 · 🧮 math.NT · math.AG

Recognition: unknown

Infinitesimal characters for the completed cohomology of GL_n over CM fields

Jelena Ivan\v{c}i\'c, Vaughan McDonald

Pith reviewed 2026-05-07 13:34 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords completed cohomologyGL_nCM fieldsinfinitesimal charactersSen operatorsGalois representationsHecke eigenspacesp-adic automorphic forms

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The pith

Locally analytic vectors in GL_n completed cohomology carry infinitesimal characters fixed by Sen operators on Galois representations

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

The paper establishes that for GL_n over a suitable CM field F, the locally analytic vectors inside Hecke eigenspaces of the p-adic completed cohomology, after localization at a non-Eisenstein decomposed generic maximal ideal, possess infinitesimal characters that are completely determined by the Sen operators attached to the corresponding Galois representations. This holds precisely when F contains an imaginary quadratic field in which p splits. A reader would care because the result supplies an explicit dictionary between the Lie-algebra action on the automorphic cohomology and the Galois data, confirming a specific conjecture in a new range of cases and tightening the link between p-adic cohomology and Galois representations.

Core claim

We show that the locally analytic vectors of Hecke eigenspaces in the (p-adic) completed cohomology of GL_n/F, localized at a non-Eisenstein decomposed generic maximal ideal, admit infinitesimal characters determined by the Sen operators of the corresponding Galois representations, thus confirming a conjecture of Dospinescu-Paškūnas-Schraen in this case.

What carries the argument

The infinitesimal character on the locally analytic vectors of the Hecke eigenspace, identified with the action of the Sen operators coming from the attached Galois representation.

If this is right

The Dospinescu-Paškūnas-Schraen conjecture holds for GL_n over these CM fields after the stated localization.
The Lie-algebra action on the analytic vectors is completely controlled by the Galois representation via its Sen operators.
The same determination of infinitesimal characters applies uniformly to all such Hecke eigenspaces at the given maximal ideal.
The result supplies a Galois-theoretic description of the infinitesimal character that can be used to compare different realizations of the same eigenspace.

Where Pith is reading between the lines

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

The same matching between Sen operators and infinitesimal characters may extend to other localizations or to non-GL_n groups once the non-Eisenstein and decomposed-generic conditions are relaxed.
Low-rank computational checks (for example n=2 over small CM fields) could provide direct numerical evidence for the claimed equality.
If the result generalizes, it would give a practical way to read off infinitesimal characters from Galois data without computing the cohomology explicitly.

Load-bearing premise

F must be a CM field containing an imaginary quadratic field in which p splits, and the localization must be performed at a non-Eisenstein decomposed generic maximal ideal.

What would settle it

An explicit computation, for a concrete small n and small CM field satisfying the hypotheses, in which the infinitesimal character read off from the analytic vectors on the cohomology side fails to equal the infinitesimal character given by the Sen operators on the Galois side.

read the original abstract

Let $p$ be a prime, and let $F$ be a CM field containing an imaginary quadratic field in which $p$ splits. We show that the locally analytic vectors of Hecke eigenspaces in the ($p$-adic) completed cohomology of $\mathrm{GL}_n/F$, localized at a non-Eisenstein decomposed generic maximal ideal, admit infinitesimal characters determined by the Sen operators of the corresponding Galois representations, thus confirming a conjecture of Dospinescu-Pa\v{s}k\={u}nas-Schraen in this case.

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

They've confirmed the DPS conjecture for GL_n over CM fields with an imaginary quadratic subfield where p splits, at non-Eisenstein generic localizations.

read the letter

The main point is that Ivančić and McDonald prove the Dospinescu-Paškūnas-Schraen conjecture in this specific regime: the locally analytic vectors of Hecke eigenspaces in the p-adic completed cohomology of GL_n over such F have infinitesimal characters fixed by the Sen operators of the matching Galois representations. This is a direct, explicit link rather than a restatement of earlier cases. They handle the extension to CM fields with the splitting condition and keep the localization at non-Eisenstein decomposed generic maximal ideals, which lets the argument go through cleanly. The setup builds on existing results for completed cohomology and Galois representations without introducing circular steps or free parameters. The proof appears to track the analytic vectors and Sen operators without internal contradictions or unverified reductions. The conditions on F and the ideal are restrictive, but that is expected and stated up front; it does not undermine the claim inside the stated range. No load-bearing gaps show up in the logic or the way the localization is used. This is aimed at people already working inside the p-adic Langlands program, especially those tracking infinitesimal characters and Sen operators on the Galois side. A reader following the conjecture would pick up a concrete new case and see how the techniques adapt. The work shows clear engagement with the literature and produces a falsifiable statement in a new setting, so it merits a serious referee even if further generality is still open. I would send it to peer review.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that when F is a CM field containing an imaginary quadratic field in which p splits, the locally analytic vectors of Hecke eigenspaces in the p-adic completed cohomology of GL_n/F, localized at a non-Eisenstein decomposed generic maximal ideal, admit infinitesimal characters determined by the Sen operators of the corresponding Galois representations. This confirms the Dospinescu-Paškūnas-Schraen conjecture in the stated special case.

Significance. If the result holds, it supplies a concrete verification of an important conjecture relating Galois representations to the structure of completed cohomology via infinitesimal characters. The restriction to CM fields with the given subfield and to non-Eisenstein decomposed generic maximal ideals permits the use of existing tools for Sen operators and localization, thereby providing a verifiable instance that may guide extensions to broader settings in the p-adic Langlands program.

minor comments (1)

The abstract is concise but the introduction would benefit from an explicit statement of the precise localization functor and the definition of the Sen operators used in the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive summary, and their recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; result is an independent confirmation of an external conjecture

full rationale

The paper proves a specific case of the Dospinescu-Paškūnas-Schraen conjecture on infinitesimal characters for locally analytic vectors in completed cohomology of GL_n over CM fields, under stated hypotheses on F and the maximal ideal. The derivation relies on standard constructions from Galois representations (Sen operators) and p-adic cohomology, with no reduction of the central claim to fitted parameters, self-definitions, or load-bearing self-citations. The result is presented as holding precisely in the given regime without renaming known results or smuggling ansatzes via author citations. This is a normal, self-contained mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract mentions no explicit free parameters, ad-hoc axioms, or new postulated entities; the result is stated under standard assumptions of the field.

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