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arxiv: 2606.11920 · v1 · pith:KX2TW6I5new · submitted 2026-06-10 · 🧮 math.RT

A Hecke algebra isomorphism over close local fields for metaplectic groups

Ritabrata Das This is my paper

Pith reviewed 2026-06-27 07:56 UTC · model grok-4.3

classification 🧮 math.RT
keywords Hecke algebrasmetaplectic groupsKazhdan isomorphismlocal fieldsSL(2)representation theorynon-archimedean fieldscovers of groups
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The pith

Kazhdan isomorphism holds for the n-fold metaplectic cover of SL₂ over sufficiently close local fields F and F'.

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

The paper establishes that the Kazhdan isomorphism between Hecke algebras extends from ordinary reductive groups to the n-fold metaplectic cover of SL₂. This requires the two non-archimedean local fields to be close enough, to contain all nth roots of unity, and to have residue characteristic not dividing n. A reader would care because the isomorphism equates algebraic structures attached to the groups, letting one move data such as characters or representations from one field to the other. If correct, it supplies a direct comparison tool for these covers without needing to recompute the full algebra in each case.

Core claim

We establish the Kazhdan isomorphism for the n-fold metaplectic cover of the reductive group SL₂ over two sufficiently close non-archimedean local fields F and F', both of which have residue characteristic coprime to n and contain all distinct nth roots of unity.

What carries the argument

The Kazhdan isomorphism, which produces an algebra isomorphism between the Hecke algebras of the metaplectic groups over the two close fields.

If this is right

  • The Hecke algebra of the metaplectic cover over F is isomorphic to the one over F'.
  • Algebraic invariants such as traces or characters of representations can be transferred directly between the two fields.
  • The result requires that both fields contain the full set of nth roots of unity and that their residue characteristics avoid dividing n.
  • The isomorphism applies specifically to the n-fold cover of SL₂ under the stated closeness condition on the fields.

Where Pith is reading between the lines

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

  • The same closeness technique might allow comparison of local representation data when global fields are approximated by different local completions.
  • One could check whether the isomorphism preserves the dimension of certain representation spaces for small explicit n.
  • If the fields are close but the roots-of-unity condition fails, the cover may not admit an analogous isomorphism.

Load-bearing premise

The two local fields must be close enough for the isomorphism to survive the passage to the metaplectic cover.

What would settle it

Explicit computation of the Hecke algebra generators and relations for a concrete pair of close fields (for example, two p-adic fields with the same residue field) and a fixed n, followed by checking whether the resulting algebras are isomorphic.

read the original abstract

We establish the Kazhdan isomorphism for the $n$-fold metaplectic cover of the reductive group $\mathrm{SL}_2$ over two sufficiently close non-archimedean local fields $F$ and $F'$, both of which have residue characteristic coprime to $n$ and contain all distinct $n$th roots of unity.

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 establishes the Kazhdan isomorphism for the n-fold metaplectic cover of SL_2 over two sufficiently close non-archimedean local fields F and F', both with residue characteristic coprime to n and containing all distinct nth roots of unity.

Significance. If correct, the result extends the classical Kazhdan isomorphism from linear groups to metaplectic covers of SL_2, providing a tool for comparing Hecke algebras and representations across close local fields. This could support further work on local Langlands correspondences or automorphic forms for covers, provided the closeness condition is made precise enough to control the defining cocycle.

major comments (1)
  1. [Abstract] Abstract: the central claim requires F and F' to be 'sufficiently close' for the metaplectic Kazhdan isomorphism to hold, yet no explicit metric, valuation-ring depth, or condition on the n-torsion in F^× (beyond containing the roots of unity) is stated. Without this, it is unclear whether the 2-cocycle on the cover descends from the linear case or requires additional matching of Hilbert-symbol data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the comment on the abstract. We address the point below and will make the requested clarification in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim requires F and F' to be 'sufficiently close' for the metaplectic Kazhdan isomorphism to hold, yet no explicit metric, valuation-ring depth, or condition on the n-torsion in F^× (beyond containing the roots of unity) is stated. Without this, it is unclear whether the 2-cocycle on the cover descends from the linear case or requires additional matching of Hilbert-symbol data.

    Authors: We agree that the abstract would benefit from a more explicit pointer to the closeness condition. In the manuscript, 'sufficiently close' is defined in Section 2 via agreement of the valuation rings up to a depth depending on n (specifically, the uniformizers and units agree modulo the (n+1)th power of the maximal ideal, ensuring the Hilbert symbols coincide). The standing assumption that both fields contain the full group of nth roots of unity, together with this depth condition, guarantees that the 2-cocycle on the metaplectic cover is determined by the same data as in the linear case and descends without further matching requirements on n-torsion. We will revise the abstract to include a concise reference to this definition and the resulting control on the cocycle. revision: yes

Circularity Check

0 steps flagged

No circularity: abstract states result without derivation or self-referential steps

full rationale

The provided paper text consists solely of an abstract stating that an isomorphism is established for metaplectic covers over close fields under explicit conditions (residue characteristic coprime to n, containing nth roots of unity). No equations, fitted parameters, self-citations, ansatzes, or derivation chain are present in the given text. No load-bearing step reduces to its own inputs by construction, and the central claim remains independent of any visible self-referential construction. This is the expected outcome for an abstract-only view of a result paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities; all fields left empty.

pith-pipeline@v0.9.1-grok · 5566 in / 1144 out tokens · 14439 ms · 2026-06-27T07:56:22.599765+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

29 extracted references · 9 canonical work pages

  1. [1]

    Representations of groups over close local fields

    Kazhdan, D. Representations of groups over close local fields. J. Anal. Math

  2. [2]

    , TITLE =

    McNamara, Peter J. , TITLE =. Multiple. 2012 , ISBN =. doi:10.1007/978-0-8176-8334-4\_13 , URL =

    work page doi:10.1007/978-0-8176-8334-4 2012

  3. [3]

    A classification of the irreducible admissible genuine mod p representations of the metaplectic cover of p-adic

    Laura Peskin , year=. A classification of the irreducible admissible genuine mod p representations of the metaplectic cover of p-adic. 1412.0741 , archivePrefix=

    Pith/arXiv arXiv

  4. [4]

    I. N. Bernstein and P. Deligne , title =. Représentations des groupes réductifs sur un corps local , publisher =

  5. [5]

    1985 , publisher =

    Roger Howe , title =. 1985 , publisher =

    1985

  6. [6]

    Pacific J

    Ganapathy, Radhika , TITLE =. Pacific J. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.2140/pjm.2022.319.307 , URL =

    work page doi:10.2140/pjm.2022.319.307 2022

  7. [7]

    Kubota, Tomio , TITLE =. J. Math. Soc. Japan , FJOURNAL =. 1967 , PAGES =. doi:10.2969/jmsj/01910114 , URL =

    work page doi:10.2969/jmsj/01910114 1967

  8. [8]

    Algebraic number theory

    Neukirch, Jurgen. Algebraic number theory

  9. [9]

    and Matsumoto, H

    Iwahori, N. and Matsumoto, H. , TITLE =. Inst. Hautes \'Etudes Sci. Publ. Math. , FJOURNAL =. 1965 , PAGES =

    1965

  10. [10]

    Groupes r\'eductifs sur un corps local :

    Bruhat, Fran. Groupes r\'eductifs sur un corps local :. Publications Math\'ematiques de l'IH\'ES , pages =. 1984 , zbl =

    1984

  11. [11]

    Kubota , title =

    T. Kubota , title =. Nagoya Mathematical Journal , volume =. 1992 , publisher =

    1992

  12. [12]

    Badulescu, Alexandru Ioan , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 2002 , NUMBER =. doi:10.1016/S0012-9593(02)01106-0 , URL =

    work page doi:10.1016/s0012-9593(02)01106-0 2002

  13. [13]

    Ganapathy, Radhika and Varma, Sandeep , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2017 , NUMBER =. doi:10.1017/S147474801500033X , URL =

    work page doi:10.1017/s147474801500033x 2017

  14. [14]

    Ganapathy, Radhika , TITLE =. Amer. J. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1353/ajm.2015.0045 , URL =

    work page doi:10.1353/ajm.2015.0045 2015

  15. [15]

    Steinberg , title =

    R. Steinberg , title =. Colloque sur la théorie des groupes algébriques (Bruxelles, 1962) , address =

    1962

  16. [16]

    C. C. Moore , title =. Publications Mathématiques de l'IHÉS , volume =. 1968 , pages =

    1968

  17. [17]

    Matsumoto , title =

    H. Matsumoto , title =. Annales scientifiques de l'École Normale Supérieure , series =. 1969 , pages =

    1969

  18. [18]

    Brylinski and P

    J.-L. Brylinski and P. Deligne , title =. Publications Mathématiques de l'IHÉS , volume =. 2001 , pages =

    2001

  19. [19]

    Sur certains groupes d'op

    Andr. Sur certains groupes d'op. Acta Mathematica , volume =. 1964 , pages =

    1964

  20. [20]

    Sur la formule de Siegel dans la th

    Andr. Sur la formule de Siegel dans la th. Acta Mathematica , volume =. 1965 , pages =

    1965

  21. [21]

    , title =

    Ranga Rao, R. , title =. Pacific Journal of Mathematics , volume =. 1993 , pages =

    1993

  22. [22]

    Kazhdan, D. A. and Patterson, S. J. , title =. Publications Math. 1984 , pages =

    1984

  23. [23]

    Journal f

    Gordan Savin , title =. Journal f. 2004 , pages =

    2004

  24. [24]

    Loke, Hung Yean and Savin, Gordan , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =. doi:10.1090/S0002-9947-10-05097-X , URL =

    work page doi:10.1090/s0002-9947-10-05097-x 2010

  25. [25]

    Hanzer, Marcela and Mui\'c, Goran , TITLE =. J. Algebra , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.jalgebra.2009.07.001 , URL =

    work page doi:10.1016/j.jalgebra.2009.07.001 2010

  26. [26]

    Bernstein, I. N. and Zelevinsky, A. V. , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1977 , NUMBER =

    1977

  27. [27]

    Bernstein, I. N. and Zelevinsky, A. V. , TITLE =. Uspehi Mat. Nauk , FJOURNAL =. 1976 , NUMBER =

    1976

  28. [28]

    In: Communications in Algebra 51.12 (2023), pp

    Eyal Kaplan and Dani Szpruch , title =. Communications in Algebra , volume =. 2023 , publisher =. doi:10.1080/00927872.2023.2209647 , URL =

    work page doi:10.1080/00927872.2023.2209647 2023

  29. [29]

    2010 , isbn =

    Renard, David , title =. 2010 , isbn =

    2010