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arxiv: 2607.00088 · v1 · pith:YRVVKSWEnew · submitted 2026-06-30 · 🧮 math.RT · math.AG· math.NT

Central isogenies and conjugacy classes in reductive groups

Sean Cotner This is my paper

Pith reviewed 2026-07-02 01:14 UTC · model grok-4.3

classification 🧮 math.RT math.AGmath.NT
keywords central isogeniesconjugacy classesreductive groupscentralizersunipotent elementsSteinberg theoremcomponent groupsétale covers
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The pith

Steinberg's description of centralizer components generalizes via central isogenies to explain non-reduced centralizers of unipotents.

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

The paper shows that Steinberg's description of the group of components of the centralizer of a semisimple element in a connected semisimple algebraic group G, as a subgroup of the fundamental group of G, extends to a broader setting. The extension uses central isogenies to account for the structure of centralizers of unipotent elements and thereby explains why those centralizers can fail to be reduced when the universal cover of G is not étale. A reader would care because this resolves an observed failure of reducedness in positive characteristic and supplies explicit tools for multiplicity calculations in moduli problems attached to Galois representations. The same framework yields concrete results on the absence of certain isomorphisms for groups such as PGL_p.

Core claim

Steinberg described the group of components of the centralizer of a semisimple element of a connected semisimple algebraic group G as a subgroup of the fundamental group of G. This description generalizes via central isogenies to explain that centralizers of unipotent elements can fail to be reduced when the universal cover of G is not étale.

What carries the argument

Central isogenies, which are isogenies whose kernels are central and which relate the component groups of centralizers across different covers of a reductive group.

If this is right

  • Generic multiplicities in the special fibers of moduli spaces of L-parameters are computed explicitly.
  • Generic multiplicities in universal deformation rings are computed explicitly.
  • There is no Springer isomorphism for PGL_p in characteristic p.

Where Pith is reading between the lines

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

  • The same central-isogeny technique may clarify reducedness questions for centralizers of other classes of elements in positive characteristic.
  • It supplies a uniform language for comparing component groups across isogenous groups, which could be tested in low-rank examples beyond PGL_p.

Load-bearing premise

The generalization starts from the known validity of Steinberg's description for semisimple elements and applies it in setups where the universal cover of G fails to be étale.

What would settle it

An explicit computation, in a group such as PGL_p in characteristic p, of the scheme-theoretic centralizer of a chosen unipotent element together with its geometric component group, checked against the prediction coming from the central isogeny.

read the original abstract

Steinberg described the group of components of the centralizer of a semisimple element of a connected semisimple algebraic group $G$ as a subgroup of the fundamental group of $G$. We show that this description can be generalized to explain the fact that centralizers of unipotent elements can fail to be reduced when the universal cover of $G$ is not \'etale. As applications, we compute generic multiplicities in the special fibers of moduli spaces of L-parameters and universal deformation rings, and we show there is no Springer isomorphism for $\mathrm{PGL}_p$ in characteristic $p$.

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 / 2 minor

Summary. The manuscript generalizes Steinberg's description of the component group π₀(C_G(s)) for semisimple elements s in a connected semisimple algebraic group G (as a subgroup of π₁(G)) to unipotent elements u via central isogenies. The central claim is that this explains why C_G(u) fails to be reduced precisely when the universal cover of G is not étale. Applications include explicit computations of generic multiplicities in the special fibers of moduli spaces of L-parameters and universal deformation rings, together with a proof that no Springer isomorphism exists for PGL_p in characteristic p.

Significance. If the central generalization is established with the required scheme-theoretic link, the work supplies a conceptual bridge between étale component-group data and infinitesimal non-reducedness phenomena for centralizers in positive characteristic. The applications to deformation rings and L-parameter moduli connect directly to arithmetic geometry and the Langlands program, while the explicit negative result on Springer isomorphisms for PGL_p furnishes a concrete obstruction with implications for nilpotent orbit theory.

major comments (2)
  1. [main theorem / abstract] The central claim (abstract and main theorem) asserts that the central-isogeny generalization of Steinberg's π₀ description explains the failure of reducedness for C_G(u). However, non-reducedness is detected by a non-zero nilradical in the coordinate ring O(C_G(u)), while component groups and isogenies control only the étale quotient. The manuscript does not supply an explicit step showing that the isogeny data produces a non-zero nilpotent element rather than merely describing the reduced subscheme or its components; this link is load-bearing for the stated explanation.
  2. [applications section] The applications to generic multiplicities in special fibers of L-parameter moduli and universal deformation rings rely on the same central-isogeny construction. Without the missing link from isogeny data to the nilradical, the multiplicity formulas cannot be justified at the scheme-theoretic level required for the special-fiber computations.
minor comments (2)
  1. [introduction] Notation for the fundamental group π₁(G) and the universal cover should be introduced with a brief reminder of the precise definition used (e.g., in the étale or fppf topology) to avoid ambiguity when the cover fails to be étale.
  2. [final application] The statement that there is 'no Springer isomorphism for PGL_p in characteristic p' would benefit from an explicit reference to the precise formulation of the Springer isomorphism being negated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments identify a need to make the scheme-theoretic connection between central isogenies and non-reducedness fully explicit. We address each point below and will revise accordingly.

read point-by-point responses

  1. Referee: [main theorem / abstract] The central claim (abstract and main theorem) asserts that the central-isogeny generalization of Steinberg's π₀ description explains the failure of reducedness for C_G(u). However, non-reducedness is detected by a non-zero nilradical in the coordinate ring O(C_G(u)), while component groups and isogenies control only the étale quotient. The manuscript does not supply an explicit step showing that the isogeny data produces a non-zero nilpotent element rather than merely describing the reduced subscheme or its components; this link is load-bearing for the stated explanation.

    Authors: We thank the referee for this observation. The main theorem establishes that the component group of C_G(u) is controlled by the fundamental group of G via the central isogeny φ:tilde G → G. When φ is not étale the infinitesimal kernel of φ intersects the preimage of u non-trivially, and the induced map on centralizers C_{tilde G}(tilde u) → C_G(u) is not an isomorphism of schemes; the coordinate ring of C_G(u) is obtained by base change along this map, so functions vanishing on the kernel generate a non-zero nilradical. While this reasoning is present in the proof of the main theorem, we agree that an explicit construction of a concrete nilpotent element would strengthen the exposition. In the revision we will insert a short lemma immediately after the main theorem that exhibits such a nilpotent explicitly from the Lie algebra of the kernel of φ. revision: yes

  2. Referee: [applications section] The applications to generic multiplicities in special fibers of L-parameter moduli and universal deformation rings rely on the same central-isogeny construction. Without the missing link from isogeny data to the nilradical, the multiplicity formulas cannot be justified at the scheme-theoretic level required for the special-fiber computations.

    Authors: The multiplicity formulas in the applications section are derived from the scheme-theoretic centralizers furnished by the main theorem. Once the explicit nilradical construction is added (as described in the response to the first comment), the same data will justify the multiplicity computations at the required scheme-theoretic level. We will add a brief paragraph in the applications section that recalls the new lemma and confirms that the special-fiber multiplicities are computed on the non-reduced schemes. revision: yes

Circularity Check

0 steps flagged

No circularity; generalization builds on external Steinberg input without self-referential reduction

full rationale

The paper explicitly takes Steinberg's description of π0(C_G(s)) for semisimple s as an established external result and states that it generalizes this description to unipotent elements via central isogenies. No equation or claim reduces by construction to a fitted parameter, self-citation chain, or renamed input; the derivation chain remains self-contained against the cited prior theorem, which is independent of the present work. The abstract and setup treat the Steinberg result as the starting point rather than deriving it internally.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no equations, parameters, or explicit axioms; ledger left empty pending full text.

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