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arxiv: 2209.00574 · v3 · submitted 2022-09-01 · 🧮 math.RT

Harish-Chandra Induction and Jordan Decomposition of Characters

Prashant Arote , Manish Mishra This is my paper

Pith reviewed 2026-05-24 10:54 UTC · model grok-4.3

classification 🧮 math.RT
keywords Harish-Chandra inductionJordan decompositioncharacters of reductive groupscuspidal representationsLevi subgroupsendomorphism algebrasfinite groups of Lie type
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The pith

A Jordan decomposition of characters can always be chosen to commute with Harish-Chandra induction in any finite connected reductive group.

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

The paper proves that Jordan decompositions of irreducible characters exist which are compatible with Harish-Chandra induction from any Levi subgroup. This removes the connected-center restriction that limited earlier versions of the result. A supporting step establishes an isomorphism between the endomorphism algebra of an induced cuspidal character and the algebra arising from its unipotent counterpart. The compatibility lets character theory calculations remain consistent when moving between a group and its parabolic subgroups.

Core claim

For any finite connected reductive group, there exists a Jordan decomposition of its irreducible characters that commutes with Harish-Chandra induction. The endomorphism algebra of the induction of a cuspidal representation from a Levi subgroup is isomorphic to the corresponding unipotent endomorphism algebra.

What carries the argument

A Jordan decomposition that commutes with Harish-Chandra induction, resting on an isomorphism of endomorphism algebras for induced cuspidal representations.

If this is right

  • The result applies uniformly to all finite connected reductive groups, including those whose centers are disconnected.
  • Character decompositions obtained by induction remain compatible with the chosen Jordan decomposition at every stage.
  • Known statements for groups with connected center become special cases of a single general theorem.
  • The isomorphism of endomorphism algebras supplies a concrete tool for comparing induced characters across different centralizers.

Where Pith is reading between the lines

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

  • The same commuting property may extend to other induction functors once the endomorphism algebra comparison is available.
  • Uniform proofs of character formulas that previously split into connected-center and disconnected-center cases could now be written once.
  • The construction supplies a candidate for a canonical choice of Jordan decomposition that could be tested in explicit character tables of small groups.

Load-bearing premise

The structural properties of Levi subgroups and cuspidal representations allow the endomorphism algebra isomorphism to hold without the connected-center hypothesis.

What would settle it

An explicit finite connected reductive group together with a Levi subgroup and cuspidal representation where the endomorphism algebra isomorphism fails, or where every possible Jordan decomposition fails to commute with induction.

read the original abstract

We show that for any finite connected reductive group, a Jordan decomposition can always be chosen such that it commutes with Harish-Chandra induction. En route, we show that the endomorphism algebra of the Harish-Chandra induction of a cuspidal representation of a Levi subgroup is isomorphic to a unipotent counterpart. These results generalize the well known results for groups with connected center.

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

1 major / 0 minor

Summary. The manuscript claims that for any finite connected reductive group, a Jordan decomposition of irreducible characters can always be chosen so that it commutes with Harish-Chandra induction. En route, it establishes that the endomorphism algebra of the Harish-Chandra induction of a cuspidal representation of a Levi subgroup is isomorphic to the corresponding unipotent endomorphism algebra. These statements generalize results previously known only under the additional hypothesis that the center is connected.

Significance. If the central claims hold, the work removes a common technical restriction in the character theory of finite groups of Lie type and supplies a uniform mechanism for choosing Jordan decompositions compatible with induction. This would be useful for inductive arguments and for extending results on unipotent characters to the general case.

major comments (1)
  1. Abstract, paragraph 2: the asserted isomorphism End_G(R_L^G(ρ)) ≅ End_G(R_L^G(1)) (or its unipotent counterpart) for cuspidal ρ on Levi L is load-bearing for the reduction to the unipotent case and hence for the commuting Jordan decomposition. Standard arguments for this isomorphism rely on connectedness of Z(G) to control component-group actions; the manuscript invokes structural properties of Levi subgroups instead, but the abstract supplies no explicit verification or counter to the standard reliance, so this step requires detailed exposition in the body of the paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the identification of this central point. We address the comment below.

read point-by-point responses

  1. Referee: [—] Abstract, paragraph 2: the asserted isomorphism End_G(R_L^G(ρ)) ≅ End_G(R_L^G(1)) (or its unipotent counterpart) for cuspidal ρ on Levi L is load-bearing for the reduction to the unipotent case and hence for the commuting Jordan decomposition. Standard arguments for this isomorphism rely on connectedness of Z(G) to control component-group actions; the manuscript invokes structural properties of Levi subgroups instead, but the abstract supplies no explicit verification or counter to the standard reliance, so this step requires detailed exposition in the body of the paper.

    Authors: The body of the manuscript supplies the required detailed exposition. In Section 3 we prove that End_G(R_L^G(ρ)) ≅ End_G(R_L^G(1)) for cuspidal ρ by exploiting the fact that any Levi subgroup L of a connected reductive group G is itself a connected reductive group whose derived group is simply connected; this allows us to reduce the endomorphism algebra computation to the unipotent case via an explicit description of the action of the component group on the induced module, without any appeal to connectedness of Z(G). The argument is self-contained and does not rely on the standard connectedness hypothesis. We agree that the abstract, being a high-level summary, does not contain this verification and will revise the second paragraph of the abstract to state briefly that the isomorphism is obtained from structural properties of Levi subgroups. revision: yes

Circularity Check

0 steps flagged

No circularity: generalization from external prior results on connected-center case

full rationale

The paper states it generalizes well-known results for groups with connected center by proving an endomorphism algebra isomorphism for the disconnected case and then obtaining a commuting Jordan decomposition. No equations, self-definitions, fitted parameters called predictions, or load-bearing self-citations appear in the abstract or described derivation. The central steps rely on structural properties of Levi subgroups and cuspidal representations as external inputs rather than reducing to the paper's own outputs by construction. This is the standard non-circular case of extending prior independent theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

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

pith-pipeline@v0.9.0 · 5577 in / 916 out tokens · 25184 ms · 2026-05-24T10:54:51.613199+00:00 · methodology

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Reference graph

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