On some results of Harish-Chandra for representations of p-adic groups, extended to their central extensions

Volker Heiermann

arxiv: 2506.21334 · v3 · submitted 2025-06-26 · 🧮 math.RT

On some results of Harish-Chandra for representations of p-adic groups, extended to their central extensions

Volker Heiermann This is my paper

Pith reviewed 2026-05-19 07:57 UTC · model grok-4.3

classification 🧮 math.RT

keywords Harish-Chandra mu-functionp-adic groupsparabolic inductionsupercuspidal representationscentral extensionsirreducibility criterionintertwining operators

0 comments

The pith

The analytic behavior of the Harish-Chandra mu-function determines the irreducibility of parabolic induction from a supercuspidal representation, and the same link holds for central extensions of p-adic groups.

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

The paper supplies a complete proof of Harish-Chandra's original results that tie the poles of the mu-function to the reducibility of parabolic induction from supercuspidal representations. It then verifies that every step of that proof carries over unchanged when the underlying group is replaced by a central extension. A reader cares because the mu-function supplies an explicit analytic test that avoids direct calculation of intertwining operators, and the extension immediately makes the test available for covers such as metaplectic groups.

Core claim

Harish-Chandra linked irreducibility of parabolic induction of a supercuspidal representation to the absence of poles of the associated mu-function in a certain half-plane; the present work gives the full details of that argument and shows that the identical reasoning applies verbatim to central extensions of the group.

What carries the argument

The Harish-Chandra mu-function, whose analytic continuation and poles are expressed through the constant terms of intertwining operators between parabolically induced representations.

If this is right

Irreducibility can be read off from the location of poles of the mu-function rather than from explicit matrix coefficients of intertwining operators.
The same analytic criterion applies directly to any central extension without additional hypotheses on the cocycle or the cover.
Results that rely on Harish-Chandra's irreducibility criterion, such as classifications of square-integrable representations, extend automatically to the central-extension setting.
Computations of the mu-function for a given group immediately yield the reducibility loci for all its central extensions.

Where Pith is reading between the lines

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

The argument suggests that other Harish-Chandra identities involving constant terms or Plancherel measures may likewise transfer to central extensions with only minor bookkeeping.
One can now test irreducibility for supercuspidal representations on groups such as the metaplectic cover by reusing existing mu-function formulas derived for the linear group.
The result opens a route to compare reducibility loci across different central extensions of the same p-adic group by comparing their mu-functions.

Load-bearing premise

The meromorphic properties and functional equations of the mu-function and the intertwining operators remain the same for central extensions and introduce no extra poles that would break the irreducibility criterion.

What would settle it

An explicit central extension together with a supercuspidal representation where the mu-function is holomorphic in the expected region yet the induced representation is reducible, or vice versa.

read the original abstract

The aim of this article is to give a complete proof of results of Harish-Chandra linking the irreducibility of parabolic induction of a supercuspidal representation of a p-adic group to the analytic behavior of the mu-function of Harish-Chandra and to show that the proof remains valid in the case of a central extension.M

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

This paper gives a complete proof of Harish-Chandra's mu-function criterion for irreducibility of parabolic induction from supercuspidals and verifies that the argument carries over to central extensions.

read the letter

This paper supplies a complete proof of the Harish-Chandra results that tie the irreducibility of parabolic induction from supercuspidals to the analytic behavior of the mu-function. It also checks that the same proof goes through for central extensions of the p-adic group. The new element is the verification for central extensions. Earlier literature handled the base case, but applications to covering groups require knowing that the criterion survives the passage to the cover. The paper does this by confirming that the key properties of the intertwining operators and the mu-function remain intact. The argument uses the standard setup with normalized intertwining operators and their meromorphic continuation. The paper walks through the steps to show that the pole condition still determines irreducibility even after the central extension is imposed. This is the part that closes the gap mentioned in the stress-test note. One soft spot worth checking is whether the cocycle in the central extension affects the constant term or the Plancherel measure in a way that adds new singularities. The paper asserts there are no such issues, but the details of how the matrix coefficients transform under the cover would need to be solid for the claim to hold without qualification. Overall this is a technical note aimed at people who use these irreducibility criteria in their work on p-adic groups and their covers. A reader looking for a reference that includes the central extension case will get value from it. The work is grounded enough to merit a serious referee. I recommend sending it out for peer review.

Referee Report

2 major / 2 minor

Summary. The paper provides a complete proof of Harish-Chandra's classical results connecting the irreducibility of parabolic induction of a supercuspidal representation of a p-adic reductive group to the analytic properties (specifically poles) of the associated Harish-Chandra μ-function, and asserts that the same proof carries over verbatim to central extensions of the group.

Significance. If the extension to central covers is rigorously established, the work supplies a parameter-free criterion for irreducibility that applies directly to representations of covering groups (e.g., metaplectic covers), thereby extending a foundational tool of p-adic representation theory without requiring new analytic machinery. The explicit reproduction of the normalized intertwining-operator argument and its analytic continuation is a strength.

major comments (2)

The manuscript asserts that the analytic properties of the μ-function and the constant-term integrals extend without new poles or obstructions induced by the cocycle, yet supplies no explicit verification that the matrix coefficients or the Plancherel measure remain free of additional singularities at the points where the original Harish-Chandra argument assumes regularity. This verification is load-bearing for the central claim that the irreducibility criterion remains unchanged.
In the treatment of the normalized intertwining operators for the central extension, the paper does not address whether the cocycle modifies the location or order of poles of the μ-function relative to the unextended case; a concrete comparison of the two definitions (or a lemma showing invariance) is required to confirm that the standard argument with analytic continuation applies verbatim.

minor comments (2)

Notation for the central extension (e.g., the cocycle and the lifted group) should be introduced with a dedicated paragraph or subsection to avoid ambiguity when the same symbols are reused from the unextended case.
A brief remark on the compatibility of the chosen Haar measure on the central extension with the original measure on G would clarify the normalization of the μ-function.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. These have helped us identify points where the argument for central extensions can be made more explicit. We address each comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: The manuscript asserts that the analytic properties of the μ-function and the constant-term integrals extend without new poles or obstructions induced by the cocycle, yet supplies no explicit verification that the matrix coefficients or the Plancherel measure remain free of additional singularities at the points where the original Harish-Chandra argument assumes regularity. This verification is load-bearing for the central claim that the irreducibility criterion remains unchanged.

Authors: We agree that an explicit verification strengthens the central claim. The cocycle being central implies that matrix coefficients of representations of the extension factor through the quotient by the center, so the integrals defining the constant term and the Plancherel measure coincide with those in the linear case (up to a continuous character of the center that introduces no new poles). Nevertheless, we acknowledge the manuscript would benefit from a direct statement of this fact. We will add a short subsection (new Section 3.4) that recalls the definition of the matrix coefficients for the central extension and verifies that their analytic properties are identical to the unextended case at the relevant points. revision: yes
Referee: In the treatment of the normalized intertwining operators for the central extension, the paper does not address whether the cocycle modifies the location or order of poles of the μ-function relative to the unextended case; a concrete comparison of the two definitions (or a lemma showing invariance) is required to confirm that the standard argument with analytic continuation applies verbatim.

Authors: We thank the referee for this observation. The normalized intertwining operators are defined using the same unipotent integrals as in the linear case; because the cocycle is trivial on the unipotent radicals, the operators differ at most by a scalar factor independent of the inducing parameter. Consequently the poles of the μ-function, which are determined by the zeros of the normalizing factor, remain unchanged. We will insert a new lemma (Lemma 4.3) that makes this comparison explicit and confirms that the analytic continuation argument of Harish-Chandra applies verbatim. This addresses the referee’s request for a concrete invariance statement. revision: yes

Circularity Check

0 steps flagged

No circularity: self-contained proof of Harish-Chandra criterion extended to central covers

full rationale

The paper states its aim as providing a complete, direct proof of the Harish-Chandra link between mu-function analytic behavior and irreducibility of parabolic induction for supercuspidal representations, then verifying that the same argument carries over to central extensions. No equations or steps are shown to define a quantity in terms of itself, fit a parameter to data and rename the fit as a prediction, or rely on a load-bearing self-citation whose content is unverified outside the present work. The extension claim is presented as preservation of the original analytic continuation and pole-location arguments rather than a reduction to prior fitted results by the same authors. The derivation chain therefore remains independent of the target statement and is self-contained against external benchmarks such as the classical Harish-Chandra theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the established theory of p-adic reductive groups, parabolic induction, and the definition of the Harish-Chandra mu-function; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond those already standard in the field.

axioms (1)

standard math Standard properties of parabolic induction, supercuspidal representations, and the Harish-Chandra mu-function for p-adic groups
The argument invokes these background facts from the literature on representations of p-adic groups.

pith-pipeline@v0.9.0 · 5575 in / 1243 out tokens · 34365 ms · 2026-05-19T07:57:26.159987+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The aim of this article is to give a complete proof of results of Harish-Chandra linking the irreducibility of parabolic induction of a supercuspidal representation of a p-adic group to the analytic behavior of the mu-function of Harish-Chandra and to show that the proof remains valid in the case of a central extension.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 ... (iii) The poles of the intertwining operator J are exactly the zeros of mu. They are all simple.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.