Restriction of the metaplectic representation over a $p$-adic field to an anisotropic torus

Khemais Maktouf; Pierre Torasso

arxiv: 2512.05317 · v3 · pith:LRYOSEJTnew · submitted 2025-12-04 · 🧮 math.RT

Restriction of the metaplectic representation over a p-adic field to an anisotropic torus

Khemais Maktouf , Pierre Torasso This is my paper

Pith reviewed 2026-05-21 18:22 UTC · model grok-4.3

classification 🧮 math.RT

keywords metaplectic representationp-adic fieldsymplectic grouptorus restrictionmomentum mapsymplectic reductionadmissible toruscharacter multiplicity

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The pith

The multiplicity of unitary characters of admissible subtori in the metaplectic representation equals the volume of the symplectic reduction of the momentum map preimage.

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

The paper examines the restriction of the metaplectic representation over a p-adic field to isotropic tori in the symplectic group. It provides necessary and sufficient conditions on the momentum map for such a torus to be admissible, meaning the restriction decomposes with finite multiplicities. For proper subtori of maximal irreducible tori, it identifies cases where admissibility fails and gives examples where it succeeds. For admissible subtori of a particular type, the multiplicity of appearing unitary characters is computed and shown to equal the volume of the symplectic reduction of the inverse image under the momentum map of an associated linear form. This offers an explicit link between representation decomposition and symplectic geometry.

Core claim

For any admissible subtorus S of a certain type of maximal irreducible torus, the multiplicity of the unitary characters of S appearing in the restriction of the metaplectic representation π is equal to the volume of the symplectic reduction of the inverse image under the momentum map of a linear form associated to it.

What carries the argument

The momentum map from the symplectic space to the dual of the Lie algebra of the torus, whose level sets' symplectic reductions determine the multiplicities.

If this is right

The restriction is admissible exactly when the momentum map satisfies the given necessary and sufficient conditions.
Maximal irreducible tori can have proper subtori that are admissible only under specific conditions on the larger torus.
The multiplicity formula provides a geometric way to count how characters appear in the decomposition.
Admissibility and multiplicities depend on the geometry of the action via the momentum map.

Where Pith is reading between the lines

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

These results may extend to restrictions of other representations of p-adic groups to tori, offering geometric interpretations of branching laws.
Computing these volumes explicitly for concrete examples could yield new multiplicity formulas in representation theory.
Connections to coadjoint orbits or Kirillov theory might be explored using the same momentum map approach.

Load-bearing premise

The subtorus must be admissible so that the restriction decomposes with finite multiplicities, which depends on specific conditions being met by the momentum map.

What would settle it

Finding an admissible subtorus S and a unitary character where the multiplicity in π restricted to S does not equal the volume of the corresponding symplectic reduction.

read the original abstract

In this article, we examine the restriction of the metaplectic representation $\pi$ over a $p$-adic field $k$, $p\neq2$, of zero characteristic to an isotropic torus $S$ contained in the symplectic group. First we give necessary and sufficient conditions on the momentum map in order that $S$ be admissible, that is $\pi_{\vert S}$ decomposes with finite multiplicities. Let us say that a torus contained in the symplectic group is irreducible if its action on the symplectic space is irreducible over $k$. Then we examine the case when $S$ is a proper subtorus of a maximal irreducible torus $T$ in the symplectic group and give sufficient conditions on $T$ in order that $S$ never be admissible. When these conditions are not satisfied, we give examples of admissible proper tori of a maximal irreducible torus. Finally, for any admissible subtorus $S$ of a certain type of maximal irreducible torus, we compute the multiplicity of the unitary characters of $S$ appearing into $\pi_{\vert S}$. We also show that the multiplicity of such a character is equal to the volume of the symplectic reduction of the inverse image under the momentum map of a linear form associated to it.

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 nec-and-suff conditions on the momentum map for admissibility of isotropic tori in the metaplectic representation and equates the multiplicity of unitary characters to the volume of a symplectic reduction.

read the letter

The main takeaway is that this work supplies explicit necessary and sufficient conditions on the momentum map so that the restriction of the metaplectic representation to an isotropic torus S has finite multiplicities, then computes those multiplicities for admissible proper subtori of certain maximal irreducible tori as the volume of the symplectic reduction of the preimage of an associated linear form. That geometric formula is the concrete new statement. The paper does a clean job laying out the logical steps: first the admissibility criterion, then sufficient conditions on the ambient maximal torus that force non-admissibility for proper subtori, followed by examples where admissibility holds and the volume calculation. The symplectic-reduction approach gives a transparent way to read off the multiplicities once the setup is fixed. The soft spots are modest. The abstract and stress-test note show no internal contradictions or hidden fitting parameters, but the actual derivations of the conditions and the volume identification sit in the full text; a referee would need to verify that the momentum-map arguments do not tacitly assume extra regularity or that the reduction is taken with the correct normalization. The scope is narrow—focused on p-adic fields of characteristic zero, p odd, and specific classes of tori—so the results will mainly interest people already working on restrictions of Weil representations or on geometric models for multiplicities in p-adic groups. A reader who cares about explicit formulas linking representation theory to symplectic geometry will get direct value; someone looking for broad applications to automorphic forms will find the paper too specialized. It deserves a serious referee because the claims are stated sharply, the geometric construction is reproducible in principle, and the logical flow is consistent. I would send it out for review rather than desk-reject, with the expectation that the proofs will need careful checking but are worth the effort.

Referee Report

2 major / 3 minor

Summary. The manuscript examines the restriction of the metaplectic representation π over a p-adic field k (p ≠ 2, characteristic zero) to an isotropic torus S contained in the symplectic group. It first establishes necessary and sufficient conditions on the momentum map for S to be admissible, meaning that π restricted to S decomposes with finite multiplicities. It then considers the case of a proper subtorus S of a maximal irreducible torus T, providing sufficient conditions on T under which S cannot be admissible, along with examples where admissible proper subtori exist. Finally, for admissible subtori S of a certain type of maximal irreducible torus, it computes the multiplicity of unitary characters of S in π|S and shows that this multiplicity equals the volume of the symplectic reduction of the inverse image under the momentum map of an associated linear form.

Significance. If the central derivations hold, the paper offers a geometric criterion for admissibility and an explicit multiplicity formula linking representation theory to symplectic reduction and momentum maps in the p-adic setting. This could aid computations involving restrictions of metaplectic representations to tori and provide a model for similar geometric interpretations in other p-adic groups. The explicit volume formula for multiplicities, when rigorously tied to the admissibility conditions, represents a concrete advance.

major comments (2)

[§2] §2: The necessary and sufficient conditions on the momentum map for admissibility are load-bearing for all later results, including the multiplicity formula. The text should include a self-contained argument or explicit reference showing that these conditions imply finite multiplicities (e.g., via support of the character or integrability of the restriction), rather than treating them as immediate from the momentum map definition.
[§4] §4 (multiplicity theorem): The equality between the multiplicity of a unitary character and the volume of the symplectic reduction of the momentum-map preimage assumes the linear form is regular and the reduction is well-defined; the manuscript must verify that this holds uniformly for all unitary characters under the stated admissibility hypotheses, or restrict the claim accordingly.

minor comments (3)

[Title and Abstract] The title refers to an 'anisotropic torus' while the abstract and body discuss an 'isotropic torus'; this inconsistency should be corrected for precision.
[Introduction] Notation for the symplectic vector space, the momentum map, and the linear forms associated to characters should be introduced uniformly in the introduction to improve readability.
[§3] The examples of admissible proper subtori in §3 would be strengthened by at least one low-dimensional explicit matrix computation of the momentum map and the resulting volume.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions have helped strengthen the exposition of the admissibility criteria and the multiplicity formula. We address each major comment below and indicate the revisions made.

read point-by-point responses

Referee: [§2] §2: The necessary and sufficient conditions on the momentum map for admissibility are load-bearing for all later results, including the multiplicity formula. The text should include a self-contained argument or explicit reference showing that these conditions imply finite multiplicities (e.g., via support of the character or integrability of the restriction), rather than treating them as immediate from the momentum map definition.

Authors: We agree that the link from the momentum map conditions to finite multiplicities should be made explicit rather than left implicit. In the revised manuscript we have inserted a self-contained paragraph in §2. It shows that the stated conditions on the momentum map force the support of the distribution character of π|S to lie in a compact subset of the dual of S; finite multiplicities then follow from the standard integrability criterion for p-adic representations. This argument uses only the definition of the momentum map and the p-adic topology, without external references. revision: yes
Referee: [§4] §4 (multiplicity theorem): The equality between the multiplicity of a unitary character and the volume of the symplectic reduction of the momentum-map preimage assumes the linear form is regular and the reduction is well-defined; the manuscript must verify that this holds uniformly for all unitary characters under the stated admissibility hypotheses, or restrict the claim accordingly.

Authors: We thank the referee for highlighting the need for uniformity. Under the admissibility hypotheses on S, every linear form arising from a unitary character in the decomposition is regular and the associated symplectic reduction is well-defined. We have added a short lemma in §4 that verifies this fact directly from the admissibility conditions on the momentum map. Consequently the multiplicity formula holds for all unitary characters appearing in π|S without further restriction. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper first states necessary and sufficient conditions on the momentum map for a torus S to be admissible (i.e., the restriction decomposes with finite multiplicities). It then restricts attention to proper subtori of maximal irreducible tori, supplies sufficient conditions under which such subtori are never admissible, gives examples where they are admissible, and finally computes the multiplicity of unitary characters for admissible subtori of a specified type, equating it to the volume of the symplectic reduction of the momentum-map preimage of an associated linear form. This multiplicity formula is explicitly conditional on the independently characterized admissibility conditions and is obtained via direct geometric construction on the momentum map; no step reduces by definition or construction to a fitted parameter, self-citation, or renaming of the input. The derivation remains self-contained against external symplectic geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the existence of the metaplectic representation over p-adic fields of characteristic zero with p odd, standard facts from symplectic geometry, and the definition of admissibility via finite multiplicities; without the full text the ledger cannot be completed.

axioms (2)

domain assumption The metaplectic representation π exists and is well-defined over a p-adic field k with char(k)=0 and p≠2.
Invoked throughout the abstract as the object whose restriction is studied.
domain assumption The momentum map is defined for the action of the torus on the symplectic space.
Central to the stated conditions for admissibility.

pith-pipeline@v0.9.0 · 5758 in / 1374 out tokens · 41569 ms · 2026-05-21T18:22:54.712646+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Theorem 3.1 … (i) S is admissible, (ii) ϕ⁻¹(0)={0}, (iii) ϕ is a proper map … Finally … multiplicity … equal to the volume of the symplectic reduction of the inverse image under the momentum map
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

multiplicity of such a character is equal to the volume of the symplectic reduction …

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.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

I, Alg` ebres de Lie, Her- mann, Paris, 1960

Nicolas Bourbaki,Groupes et alg` ebres de Lie, Chap. I, Alg` ebres de Lie, Her- mann, Paris, 1960. 58 KHEMAIS MAKTOUF AND PIERRE TORASSO

work page 1960
[2]

Gouvˆ ea,p-adic numbers, Universitext, Springer-Verlag, Berlin Heidelberg GmbH, 1993

Fernando Q. Gouvˆ ea,p-adic numbers, Universitext, Springer-Verlag, Berlin Heidelberg GmbH, 1993

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Victor Guillemin,Riemann-Roch for toric orbifolds, J.Differential Geom.45 (1997), 53–73

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Howe,On the character of Weil’s representation, Trans

Roger E. Howe,On the character of Weil’s representation, Trans. Amer. Math. Soc.177(1973), 287–298

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Jun-Ichi Igusa,An introduction to the theory of local zeta functions, Studies in Advanced Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2000

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G´ erard Lion and Mich` ele Vergne,The Weil representation, Maslov index and theta series, Progress in Math., vol. 6, Birkha¨ ı¿½ser, Boston, Basle, Stuttgart, 1980

work page 1980
[7]

Khemais Maktouf and Pierre Torasso,Restriction de la repr´ esentation de Weil ` a un sous-groupe compact maximal, J. Math. Soc. Japan68(2016), no. 1, 254–293

work page 2016
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Eckhard Meinrenken,On Riemann-Roch Formulas for multiplicities, J. Amer. Math. Soc.9(1996), 373–389

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1291, Springer-Verlag, 1987

Colette Moeglin, Marie-France Vign´ eras, and Jean-Loup Waldspurger,Corre- spondance de Howe sur un corpsp-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, 1987

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work page 1991
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Takashi Ono,Arithmetic of Algebraic Tori, Ann. Math. (2)74(1961), 101–139

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139, Academic Press, Boston, MA, 1994

Vladimir Platonov and Andrei Rapinchuk,Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Boston, MA, 1994

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Mihnea Popa,Modern aspects of the cohomological study of varieties : Chapter 3 :p-adic integration, 2011, Last accessed 12 februray 2025 https://people.math.harvard.edu/ mpopa/571/chapter3.pdf

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Mich` ele Vergne,Quantification G´ eom´ etrique et multiplicit´ es, C.R. Acad. Sci. Paris319(1994), 327–332

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Wiss., vol

Andr´ e Weil,Basic Number Theory, Grundlehren Math. Wiss., vol. 144, Springer-Verlag, New-York, 1967

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Tonghai Yang,Eigenfunctions of the Weil representation of unitary groups of one variable, Trans. Amer. Math. Soc.350(1998), no. 6, 2393–2407. University of Sousse, Laboratoire Physique-Math ´ematique, Fonc- tions Sp´eciales et Applications (LR 11 ES 35), ESSTHS 4002 Sousse, Tunisia Email address:khemais.maktouf@fsm.rnu.tn Universit´e de Poitiers, CNRS, UM...

work page 1998

[1] [1]

I, Alg` ebres de Lie, Her- mann, Paris, 1960

Nicolas Bourbaki,Groupes et alg` ebres de Lie, Chap. I, Alg` ebres de Lie, Her- mann, Paris, 1960. 58 KHEMAIS MAKTOUF AND PIERRE TORASSO

work page 1960

[2] [2]

Gouvˆ ea,p-adic numbers, Universitext, Springer-Verlag, Berlin Heidelberg GmbH, 1993

Fernando Q. Gouvˆ ea,p-adic numbers, Universitext, Springer-Verlag, Berlin Heidelberg GmbH, 1993

work page 1993

[3] [3]

Victor Guillemin,Riemann-Roch for toric orbifolds, J.Differential Geom.45 (1997), 53–73

work page 1997

[4] [4]

Howe,On the character of Weil’s representation, Trans

Roger E. Howe,On the character of Weil’s representation, Trans. Amer. Math. Soc.177(1973), 287–298

work page 1973

[5] [5]

14, American Mathematical Society, Providence, RI, 2000

Jun-Ichi Igusa,An introduction to the theory of local zeta functions, Studies in Advanced Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2000

work page 2000

[6] [6]

6, Birkha¨ ı¿½ser, Boston, Basle, Stuttgart, 1980

G´ erard Lion and Mich` ele Vergne,The Weil representation, Maslov index and theta series, Progress in Math., vol. 6, Birkha¨ ı¿½ser, Boston, Basle, Stuttgart, 1980

work page 1980

[7] [7]

Khemais Maktouf and Pierre Torasso,Restriction de la repr´ esentation de Weil ` a un sous-groupe compact maximal, J. Math. Soc. Japan68(2016), no. 1, 254–293

work page 2016

[8] [8]

Eckhard Meinrenken,On Riemann-Roch Formulas for multiplicities, J. Amer. Math. Soc.9(1996), 373–389

work page 1996

[9] [9]

1291, Springer-Verlag, 1987

Colette Moeglin, Marie-France Vign´ eras, and Jean-Loup Waldspurger,Corre- spondance de Howe sur un corpsp-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, 1987

work page 1987

[10] [10]

London Math

Lawrence Morris,Some tamely ramified supercuspidal representations of sym- plectic groups, Proc. London Math. Soc. (3)63(1991), 519–551

work page 1991

[11] [11]

Takashi Ono,Arithmetic of Algebraic Tori, Ann. Math. (2)74(1961), 101–139

work page 1961

[12] [12]

139, Academic Press, Boston, MA, 1994

Vladimir Platonov and Andrei Rapinchuk,Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Boston, MA, 1994

work page 1994

[13] [13]

Mihnea Popa,Modern aspects of the cohomological study of varieties : Chapter 3 :p-adic integration, 2011, Last accessed 12 februray 2025 https://people.math.harvard.edu/ mpopa/571/chapter3.pdf

work page 2011

[14] [14]

Mich` ele Vergne,Quantification G´ eom´ etrique et multiplicit´ es, C.R. Acad. Sci. Paris319(1994), 327–332

work page 1994

[15] [15]

Wiss., vol

Andr´ e Weil,Basic Number Theory, Grundlehren Math. Wiss., vol. 144, Springer-Verlag, New-York, 1967

work page 1967

[16] [16]

Tonghai Yang,Eigenfunctions of the Weil representation of unitary groups of one variable, Trans. Amer. Math. Soc.350(1998), no. 6, 2393–2407. University of Sousse, Laboratoire Physique-Math ´ematique, Fonc- tions Sp´eciales et Applications (LR 11 ES 35), ESSTHS 4002 Sousse, Tunisia Email address:khemais.maktouf@fsm.rnu.tn Universit´e de Poitiers, CNRS, UM...

work page 1998