The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments

Runze Wang

arxiv: 2604.09138 · v3 · submitted 2026-04-10 · 🧮 math.RT

The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments

Runze Wang This is my paper

Pith reviewed 2026-05-10 16:51 UTC · model grok-4.3

classification 🧮 math.RT

keywords Iwahori-spherical representationsGL_nK_+-fixed vectorsZelevinsky segmentsKostka numbersdominance orderp-adic groups

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The pith

For generic Iwahori-spherical GL_n representations, K_+-fixed vectors decompose into K/K_+ irreducibles exactly when their partitions are dominated by one attached to the representation, with Kostka multiplicities.

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

The paper examines the space of vectors fixed by the subgroup K_+ inside Iwahori-spherical representations of GL_n over a non-archimedean local field. For generic such representations it shows that the decomposition into irreducible modules of the finite group K/K_+ is governed by a single partition coming from the representation: an irreducible appears only when its partition is dominated by that one, and the multiplicity is then a Kostka number. For arbitrary irreducible Iwahori-spherical representations the paper attaches a partition from the Zelevinsky segment data, proves that every occurring module must have a dominated partition, shows that the module for the attached partition itself appears exactly once, and supplies a combinatorial algorithm that computes the precise list of modules and their multiplicities. The work answers a question of Prasad on the structure of these fixed-vector spaces.

Core claim

For a generic Iwahori-spherical representation, its decomposition into irreducible modules of K/K_+ is controlled by a partition determined by the representation: an irreducible module occurs only if its partition is dominated by that partition, and when it occurs the multiplicity is a Kostka number. For an arbitrary irreducible Iwahori-spherical representation, we attach a partition from its data and prove a necessary condition: any occurring module must correspond to a partition dominated by this one, and the module attached to the partition itself occurs exactly once. We also give a combinatorial algorithm which, by further computation, determines precisely which modules actually occur.

What carries the argument

The partition attached to an Iwahori-spherical representation from its Zelevinsky segment data, which controls the K_+-fixed space by the dominance partial order on partitions together with Kostka numbers as multiplicities.

If this is right

In the generic case every occurring irreducible appears with multiplicity equal to the corresponding Kostka number K_{\lambda,\mu}.
No irreducible whose partition fails to be dominated by the representation's partition can appear in the K_+-fixed space.
The combinatorial algorithm gives an explicit, finite procedure to list the exact decomposition for any given irreducible Iwahori-spherical representation.
The result supplies a bridge between the representation theory of p-adic GL_n and the combinatorics of partitions and symmetric-group modules.

Where Pith is reading between the lines

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

The space of K_+-fixed vectors may be realized as a module induced from a Young subgroup corresponding to the attached partition.
Analogous dominance-plus-Kostka controls might hold for fixed vectors under deeper congruence subgroups or for other split reductive groups.
Explicit tables of multiplicities for small n obtained by running the algorithm would give concrete test data for the general statement.

Load-bearing premise

The map that attaches a partition to an arbitrary irreducible Iwahori-spherical representation from its Zelevinsky segment data is well-defined and compatible with the dominance order in the sense that only dominated partitions can occur.

What would settle it

Pick a concrete non-generic irreducible Iwahori-spherical representation of GL_3, compute its attached partition and run the combinatorial algorithm to list all occurring modules and multiplicities; the claim fails if any module whose partition is not dominated by the attached one appears or if the multiplicity of the maximal partition is not exactly one.

read the original abstract

We study the space of K_+-fixed vectors of Iwahori-spherical representations of GL_n over a non-archimedean local field. For a generic Iwahori-spherical representation, we show that its decomposition into irreducible modules of the finite Lie group K/K_+ is controlled by a partition determined by the representation: an irreducible module occurs only if its partition is dominated by that partition, and when it occurs the multiplicity is a Kostka number. For an arbitrary irreducible Iwahori-spherical representation, we attach a partition from its data and prove a necessary condition: any occurring module must correspond to a partition dominated by this one, and the module attached to the partition itself occurs exactly once. We also give a combinatorial algorithm which, by further computation, determines precisely which modules actually occur and with what multiplicities. This answers a question of Prasad.

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

The paper gives a concrete combinatorial rule for the K_+-fixed vectors in Iwahori-spherical GL_n reps via partitions, dominance, and Kostka numbers, plus a necessary condition and algorithm for the general case.

read the letter

The main takeaway is that the paper gives an explicit combinatorial description of the K_+-fixed vectors for Iwahori-spherical representations of GL_n. In the generic case, the decomposition into irreducibles of K/K_+ is controlled by a partition associated to the representation, with occurrence only for dominated partitions and multiplicities given by Kostka numbers. For general irreducibles, they attach a partition from the Zelevinsky segment data, prove that only dominated partitions can occur, and supply a combinatorial algorithm to find the exact multiplicities. This answers Prasad's question directly with something computable rather than abstract existence statements. The work does a good job connecting the representation theory to partition combinatorics in a concrete way. The algorithm part is especially useful since it allows precise computation rather than just existence. A possible soft spot is the step where a partition is attached to an arbitrary irreducible Iwahori-spherical representation. The abstract says this is done from the segment data and that the dominance condition holds, but I would check whether the attachment is uniquely defined and always compatible with the dominance partial order. If there is any ambiguity there, the necessary condition might not be as clean. The lack of examples in the abstract also means the full paper needs to include some to illustrate the algorithm. This is aimed at researchers in the representation theory of p-adic groups, specifically those dealing with Iwahori-spherical representations and questions about fixed vectors under subgroups like K_+. A reader who knows the Zelevinsky classification would get the most out of it. The paper shows clear thinking on the combinatorial side and engages with the literature, so it deserves a serious referee to go over the details and proofs. I would recommend sending it for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the space of K_+-fixed vectors in Iwahori-spherical representations of GL_n over a non-archimedean local field. For generic representations it shows that the decomposition into irreducible K/K_+-modules is governed by a partition attached to the representation: an irreducible occurs only if its partition is dominated by this one, with multiplicity equal to a Kostka number. For arbitrary irreducible Iwahori-spherical representations the paper attaches a partition to the Zelevinsky segment data, proves that any occurring module must have partition dominated by this one, shows that the module for the attached partition occurs with multiplicity one, and supplies a combinatorial algorithm that computes the precise multiplicities. The work answers a question of Prasad.

Significance. If the claims hold, the results give an explicit combinatorial description of the K_+-fixed vectors in terms of Zelevinsky segments, dominance order on partitions, and Kostka numbers. This supplies a practical tool for computing multiplicities in the finite-group decomposition and directly resolves an open question, potentially aiding further work on branching rules or fixed-vector problems in p-adic representation theory.

minor comments (3)

[§2–3] The explicit rule attaching a partition to the Zelevinsky segment data for non-generic representations should be stated as a numbered definition or algorithm early in the text (e.g., before the statement of the necessary-condition theorem) to make the dominance compatibility immediately verifiable.
[§5] A small table or worked example illustrating the combinatorial algorithm for a non-generic representation of small rank (say n=3 or 4) would clarify the steps and help readers check the multiplicity claims.
[Introduction] The notation K/K_+ is introduced without a brief reminder of the standard Iwahori subgroup K and its pro-unipotent radical K_+; adding one sentence in the introduction would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: theorems on standard objects with independent combinatorial content

full rationale

The paper states new results about the K_+-fixed vectors of Iwahori-spherical GL_n-representations, attaching a partition to Zelevinsky segment data and proving dominance and multiplicity statements as theorems rather than by definition or tautology. The generic case uses Kostka numbers as a known combinatorial invariant, while the arbitrary case supplies an explicit necessary condition plus a separate algorithm for exact multiplicities; neither reduces to a fitted parameter or self-referential input. The attachment rule and order-compatibility are presented as proven consequences of the standard Zelevinsky classification, which is external to the paper. No load-bearing self-citation or ansatz smuggling appears in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the Zelevinsky classification of irreducible representations of GL_n, the definition of Iwahori-spherical representations, and standard facts about the finite group K/K_+ and its irreducibles labeled by partitions. No new entities are introduced.

axioms (2)

domain assumption Every irreducible Iwahori-spherical representation is classified by a multiset of Zelevinsky segments.
Invoked when attaching a partition to the representation data.
domain assumption The dominance partial order on partitions governs the occurrence of irreducibles of K/K_+ in the fixed-vector space.
Central to the necessary condition stated in the abstract.

pith-pipeline@v0.9.0 · 5449 in / 1481 out tokens · 44622 ms · 2026-05-10T16:51:45.484508+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

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Bernstein, I. N., and Zelevinsky, A. V.Representations of the groupGL(n, F) whereFis a non-Archimedean local field. Russian Mathematical Surveys, 31(3):1–68, 1976

work page 1976
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N., and Zelevinsky, A

Bernstein, I. N., and Zelevinsky, A. V.Induced representations of reduc- tivep-adic groups. I. Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure, 10(4):441–472, 1977. 22

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[3]

Borel,Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent

A. Borel,Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math.35(1976), 233–259

work page 1976
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Casselman, W.The unramified principal series ofp-adic groups. I. The spher- ical function. Compositio Mathematica, 40(3):387–406, 1980

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Y., and Savin, G.Iwahori component of the Gelfand–Graev repre- sentation

Chan, K. Y., and Savin, G.Iwahori component of the Gelfand–Graev repre- sentation. Math. Z. 288 (2018), no. 1-2, 125–133

work page 2018
[6]

Geck and N

M. Geck and N. Jacon,Representations of Hecke Algebras at Roots of Unity, Springer, 2011

work page 2011
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London Mathematical Society Monographs, New Series, 21

Geck, M., and Pfeiffer, G.Characters of Finite Coxeter Groups and Iwahori- Hecke Algebras. London Mathematical Society Monographs, New Series, 21. Oxford University Press, 2000

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[8]

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Moy and G

A. Moy and G. Prasad,Unrefined minimalK-types forp-adic groups, Invent. Math.116(1994), 393–408

work page 1994
[10]

Manuscripta Mathematica, 152(1–2):241–245, 2017

Mishra, M., and R¨ osner, M.Genericity under parahoric restriction. Manuscripta Mathematica, 152(1–2):241–245, 2017

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[11]

arXiv:2511.06438v1 [math.RT], November 2025

Prasad, D.Some questions about representations ofp-adic groups. arXiv:2511.06438v1 [math.RT], November 2025

work page arXiv 2025
[12]

Publications Math´ ematiques de l’IH´ES, 85(1):97–191, 1997

Schneider, P., and Stuhler, U.Representation theory and sheaves on the Bruhat-Tits building. Publications Math´ ematiques de l’IH´ES, 85(1):97–191, 1997

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[13]

arXiv:2603.22931 [math.RT], 2026

Wang, R.Classify all representations which contain a Steinberg in their hy- perspecial subgroup. arXiv:2603.22931 [math.RT], 2026

work page arXiv 2026
[14]

V.Induced representations of reductivep-adic groups

Zelevinsky, A. V.Induced representations of reductivep-adic groups. II. On irreducible representations ofGL(n). Annales Scientifiques de l’´Ecole Normale Sup´ erieure, 13(2):165–210, 1980. 23

work page 1980

[1] [1]

N., and Zelevinsky, A

Bernstein, I. N., and Zelevinsky, A. V.Representations of the groupGL(n, F) whereFis a non-Archimedean local field. Russian Mathematical Surveys, 31(3):1–68, 1976

work page 1976

[2] [2]

N., and Zelevinsky, A

Bernstein, I. N., and Zelevinsky, A. V.Induced representations of reduc- tivep-adic groups. I. Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure, 10(4):441–472, 1977. 22

work page 1977

[3] [3]

Borel,Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent

A. Borel,Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math.35(1976), 233–259

work page 1976

[4] [4]

Casselman, W.The unramified principal series ofp-adic groups. I. The spher- ical function. Compositio Mathematica, 40(3):387–406, 1980

work page 1980

[5] [5]

Y., and Savin, G.Iwahori component of the Gelfand–Graev repre- sentation

Chan, K. Y., and Savin, G.Iwahori component of the Gelfand–Graev repre- sentation. Math. Z. 288 (2018), no. 1-2, 125–133

work page 2018

[6] [6]

Geck and N

M. Geck and N. Jacon,Representations of Hecke Algebras at Roots of Unity, Springer, 2011

work page 2011

[7] [7]

London Mathematical Society Monographs, New Series, 21

Geck, M., and Pfeiffer, G.Characters of Finite Coxeter Groups and Iwahori- Hecke Algebras. London Mathematical Society Monographs, New Series, 21. Oxford University Press, 2000

work page 2000

[8] [8]

Kaletha and G

T. Kaletha and G. Prasad,Bruhat–Tits Theory: A New Approach, New Math. Monogr.44, Cambridge Univ. Press, 2023

work page 2023

[9] [9]

Moy and G

A. Moy and G. Prasad,Unrefined minimalK-types forp-adic groups, Invent. Math.116(1994), 393–408

work page 1994

[10] [10]

Manuscripta Mathematica, 152(1–2):241–245, 2017

Mishra, M., and R¨ osner, M.Genericity under parahoric restriction. Manuscripta Mathematica, 152(1–2):241–245, 2017

work page 2017

[11] [11]

arXiv:2511.06438v1 [math.RT], November 2025

Prasad, D.Some questions about representations ofp-adic groups. arXiv:2511.06438v1 [math.RT], November 2025

work page arXiv 2025

[12] [12]

Publications Math´ ematiques de l’IH´ES, 85(1):97–191, 1997

Schneider, P., and Stuhler, U.Representation theory and sheaves on the Bruhat-Tits building. Publications Math´ ematiques de l’IH´ES, 85(1):97–191, 1997

work page 1997

[13] [13]

arXiv:2603.22931 [math.RT], 2026

Wang, R.Classify all representations which contain a Steinberg in their hy- perspecial subgroup. arXiv:2603.22931 [math.RT], 2026

work page arXiv 2026

[14] [14]

V.Induced representations of reductivep-adic groups

Zelevinsky, A. V.Induced representations of reductivep-adic groups. II. On irreducible representations ofGL(n). Annales Scientifiques de l’´Ecole Normale Sup´ erieure, 13(2):165–210, 1980. 23

work page 1980