What are the extended pure inner forms of a cover?

Luozi Shi; Yifei Zhao

REVIEW 2 major objections 3 minor 22 references

What are the extended pure inner forms of a cover?

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that every L-parameter $\sigma$ has a basic isocrystal $\beta$ for which Weissman's obstruction vanishes, so the empty L-packets of a cover reappear on an extended pure inner form $\tilde G_\beta$.

desk verdict A strong, serious framework paper on extended pure inner forms for covers, with a real but likely fixable gap in the torus duality proof that the authors flag themselves. read the letter →

arxiv 2506.08696 v1 pith:P2L6TRDG submitted 2025-06-10 math.RT math.NT

Luozi Shi , Yifei Zhao This is my paper · ORCID

classification math.RTmath.NT MSC 22E5011F7020G25

keywords extendedpureinnerformsétalemetaplecticcoverslocalLanglandscorrespondenceL-packetsisocrystalscoveringgroupsWeissmanobstructioncentralcorecharacters

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The proposal to study all extended pure inner forms together in the local Langlands correspondence is carried over to covering groups. For each basic $G$-isocrystal $\$\beta$$, the paper constructs a cover $\tilde G_\$\beta$$ of the extended pure inner form $G_\$\beta$(F)$ from a single étale metaplectic cover of $G$, and it proves that Weissman's obstruction $\Omega_\$\beta$(\$\sigma$)$ is governed by the Kottwitz invariant: it vanishes exactly when $\gamma(\mathrm{Kott}(\$\beta$)) = \$\Omega$(\$\sigma$)^{-1}$. Since the map $\gamma$ is surjective, every L-parameter has at least one basic $\$\beta$$ with vanishing obstruction, and the possible $\$\beta$$ form a torsor under the image of $(\pi_1 G^\sharp)^{\mathrm{Gal}_F}$. Assuming the conjectural compatibility with central core characters, this shows that Weissman's empty L-packets are not missing from the whole family of extended pure inner forms; they appear on a nontrivial $\tilde G_\$\beta$$. For tori the paper makes this unconditional: the fiber of the correspondence is finite and nonempty precisely when the obstruction vanishes.

What carries the argument

The load-bearing object is the canonical quadratic structure of an étale metaplectic cover (Proposition 3.1.3): for any $G$-torsor $E$ and any $Z$-torsor $\mathcal Z$, the difference $a^*\mu - p_1^*\mu - p_2^*\mu_Z$ is canonically isomorphic to $b_2 \otimes B\$Psi^{{\otimes 2}}$(E,\mathcal Z)$, a formula whose classical shadow is the Chern-class identity $c_2(E\otimes L)-c_2(E)-c_2($L^{{\oplus n}}$)=(n-1)c_1(\det E)\cup c_1(L)$. This quadratic structure yields the $BZ^\sharp$-equivariance of $\mu$, identifies the pullback cover $\tilde Z_\$\beta$$ as $\tilde Z + \int_F (b_2\otimes B\$Psi^{{\otimes 2}}$)(\$\beta$,\cdot)$, and, through Tate duality and the Kottwitz invariant, turns the ratio of central core characters into the surjection $\gamma$. A second ingredient is the splitting of $\mu_{G^\sharp_{\mathrm{ab}}}$ into a 2-torsion “sign” component and a $\mathbb Z$-linear component; the sign component is matched with the Hilbert-symbol cover and the meta-Weil group, and the $\mathbb Z$-linear component is handled by Langlands duality for tori.

What would settle it

Take $F=\mathbb Q_p$ with $p$ odd, let $G=\mathrm{GL}_2$ with the central extension from Remark 4.5.4, and choose a nontrivial $\mathbb Z$-isocrystal $\$\beta$$. Proposition 4.5.3 predicts that $\tilde G_\$\beta$$ differs from $\tilde G$ by the cover $\int_F(b_2\otimes B\$Psi^{{\otimes 2}}$)(\cdot,\$\beta$)$, while Theorem 4.3.9 predicts that the central-core character on $K$ is multiplied by $\gamma(\mathrm{Kott}(\$\beta$))$; computing this character directly by evaluating the extension class on $K$ would confirm or refute the criterion, and a single mismatch would falsify Theorem B(2).

Watch

Extended reading notes

Core claim

The paper's central result, Theorem B(2), is an exact criterion for when Weissman's obstruction to an L-packet vanishes. For any $G$-isocrystal $\$\beta$$ and any L-parameter $\$\sigma$$, $\Omega_\$\beta$(\$\sigma$) = 1$ if and only if $\gamma(\mathrm{Kott}(\$\beta$)) = \$\Omega$(\$\sigma$)^{-1}$, where $\mathrm{Kott}$ is the Kottwitz invariant and $\gamma$ is the surjection $(\pi_1 G)^{\mathrm{Gal}_F} \to \mathrm{Hom}(K, \mathbb{C}^\times)$ built from the quadratic form attached to the cover. Since $\gamma$ is surjective, for each $\$\sigma$$ there is always at least one basic $\$\beta$$ satisfying the equality, and the collection of such $\$\beta$$ is a torsor under the image of $(\pi_1 G^\sharp)^{\mathrm{Gal}_F}$. The authors interpret this as saying that the empty packets Weissman found for a single cover are not fundamentally missing: they occur on some extended pure inner form $\tilde G_\$\beta$$, provided the conjectural local Langlands correspondence for covers is compatible with central core characters. For tori the result is unconditional: Proposition 4.4.4 shows $\mathrm{LLC}_\$beta^{{-1}}$(\$\sigma$)$ is finite and nonempty exactly when the same equality holds.

Load-bearing premise

The load-bearing premise is the imported construction that converts an étale metaplectic cover of an algebraic group into a topological cover of its F-points; every cover $\tilde G_\$\beta$$ is built from it, and the nonempty-packet conclusion additionally assumes the still-conjectural compatibility of the local Langlands correspondence with central core characters.

Editorial extensions

If this is right

For every L-parameter $\sigma$, at least one basic $G$-isocrystal $\beta$ satisfies $\Omega_\beta(\sigma)=1$; if the conjectural compatibility (4.27) holds, the fiber $\mathrm{LLC}_\beta^{-1}(\sigma)$ is nonempty, so no parameter is empty across the whole family of extended pure inner forms.
The set of basic classes $\beta$ with vanishing obstruction is a torsor under the image of $(\pi_1 G^\sharp)^{\mathrm{Gal}_F}$, so the trivial form need not be among them; the quotient $(\pi_1 G)^{\mathrm{Gal}_F}/(\pi_1 G^\sharp)^{\mathrm{Gal}_F}$ measures how far a single cover's packet is from being full.
For tori, the local Langlands correspondence is constructed for every $\tilde T_\beta$, and its fibers are finite and nonempty exactly when $\gamma(\mathrm{Kott}(\beta)) = \Omega(\sigma)^{-1}$.
Even if the input is a Brylinski–Deligne (K-theory) cover, its translate by a nontrivial isocrystal generally is not one; the natural home for extended pure inner forms of covers is the category of étale metaplectic covers.
The parametrization by basic isocrystals via Kottwitz's bijection makes the family $\{\mathrm{LLC}_\beta\}$ a single correspondence on the set of extended pure inner forms rather than a separate problem for each inner form.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the central-core compatibility is later proved for all covers, Theorem B(2) would make the Langlands correspondence surjective after passing to the disjoint union over all basic $\beta$; this is a checkable prediction for individual groups such as $\mathrm{GL}_2$ with its rank-one metaplectic cover.
The exact formula $\Omega_\beta(\sigma)=\Omega(\sigma)\gamma(\mathrm{Kott}(\beta))$ suggests that other L-packet invariants, such as Whittaker-normalized characters or endoscopic transfers, could be twisted by the same Kottwitz-invariant factor when moving between extended pure inner forms.
Remark 4.5.4 indicates that the class of covers arising from algebraic $K$-theory is not closed under isocrystal translation; testing whether the translated Kazhdan–Patterson cover admits a genuinely new non-$K_2$ structure would clarify what the larger étale metaplectic category is needed for.
One could adapt the quadratic-structure computation to the geometric setting, where the Kottwitz invariant should be replaced by a cohomological class on a curve; a geometric analogue of the torsor conclusion would predict that metaplectic geometric Langlands parameters always live on some translated cover.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's what I'd tell you about Shi–Zhao, arXiv:2506.08696. It does something genuinely new: it takes Kottwitz's 'all extended pure inner forms together' philosophy and makes it work for étale metaplectic covers, constructing a cover G̃_β for each G-isocrystal β and defining a Weissman obstruction Ω_β(σ). Theorem B is the payoff: Ω_β(σ)=1 iff γ(Kott(β))=Ω(σ)^{-1}, so every L-parameter has a home on some basic form. Theorem A extends Weissman's toric LLC from split K-theoretic covers to all étale metaplectic tori, with an independent proof that handles the 2-torsion sign. The writing is clear given the machinery; the canonical quadratic structure in §3 is a nice piece of geometry. Credit where due: this is a real advance, not a repackaging.

The soft spots: the whole construction runs through the integral functor (1.6) imported from Zhao's earlier work [Zha22]. That is fine as a citation, but it means the paper's foundations are not self-contained. More troubling is §2.3.6. In the proof of Theorem 2.1.6 for a general torus, the authors embed T into an induced torus T′, extend µ to µ′, and then say they omit the verification that the resulting isomorphism is independent of choices. The stress-test is right that this is load-bearing: the central-core character map (2.40), and hence Ω_β, depends on the canonicality of (2.6). If independence fails, the left side of Theorem B(2) is not well-defined. I don't think the gap is fatal—it looks like a check that can be done by comparing two cocycles—but it is not cosmetic, and the paper should fix it before I would rely on Theorem B as stated. They also rely on conjectural compatibility (4.27) for the interpretation of empty packets; they flag this honestly.

Who is this for? Specialists in the Langlands program for covering groups. It deserves a serious referee: the framework is important, the main theorems are mostly proved in detail, and the gap is identifiable and presumably repairable. I'd send it to peer review, and if I were refereeing I'd ask for the independence check and for a statement of exactly which parts of [Zha22] are imported.

Referee Report

2 major / 3 minor

Summary. The paper proposes an extension of Kottwitz's extended pure inner forms to covering groups. For a reductive group F-scheme G, an étale metaplectic cover μ: BG → B^4A(1), and a G-isocrystal β, the authors construct a cover G~_β of the F-points of the associated form G_β. They prove Theorem A, a canonical Langlands duality for sharp tori, by splitting μ into a 2-torsion sign component and a Z-linear component, and use it to define central core characters. The main result, Theorem B, characterizes Weissman's obstruction Ω_β(σ): it vanishes if and only if γ(Kott(β)) = Ω(σ)^{-1}, so every L-parameter σ admits some basic G-isocrystal β with Ω_β(σ)=1. For tori this vanishing is shown equivalent to nonemptiness of the LLC fiber, conditional on Weissman's conjectural local Langlands correspondence and its compatibility with central core characters.

Significance. If the proofs are completed, this is a substantive contribution: it gives a concrete mechanism by which Weissman's empty L-packets can be populated on extended pure inner forms, connects Kaletha's and Weissman's frameworks, and provides an explicit torsor statement in Theorem B. The paper is honest about its inputs: the foundational functor (1.6) is imported from [Zha22], the LLC and the compatibility diagram (4.27) are labeled conjectural, and the algebraic criterion (4.28) is unconditional and checkable. The treatment of the sign component in §2.2 is detailed and includes an independent proof of Weissman's torus case. However, two load-bearing steps in the proof of Theorem A are not justified as written; these must be repaired before the central claims can be taken as established.

major comments (2)

[§2.3.6] The proof of Theorem 2.1.6 for general tori uses a reduction to an induced torus T′ via an arbitrary extension μ′ of μ. The authors state: 'We omit the verification that this isomorphism is independent of the choice of F1 and the extension μ′.' This is not a cosmetic omission. The isomorphism (2.28) for T is obtained by functoriality from T′, and two different extensions can differ by a class in H^2(Spec F, Hom(Λ′/Λ, A)), which at the level of covers corresponds to a character of T(F). A change of this character alters the bijection (2.7), hence the central-core character map (2.40) and the definition of Ω_β in §4.1.7 and §4.3.10. The left-hand side of Theorem B(2) is therefore not well-defined until this independence is proved. Please supply the missing verification or replace the reduction with a choice-independent argument.
[§2.3.3, Lemma 2.3.4] The proof of Lemma 2.3.4 asserts that the colimit of H^2(Spec F, A_1) over finite subgroups A_1 of C^× containing A vanishes, and uses this to identify the colimit of ∗/Γ(Spec F, B A_1) with Γ(Spec F, B^2 A_1). For a non-archimedean local field this colimit is generally nonzero: for F=Q_5 and A_1=μ_4, H^2(F,Z/4) ≅ Z/4, and the directed system over p′-power roots of unity stabilizes at Z/(q−1), not at zero. Thus the reduction to the neutral component is not justified. Since Lemma 2.3.4 supplies the split-torus case of the Z-linear component of Theorem 2.1.6, the proof of Theorem A is incomplete at this point. The authors should either correct the colimit statement or prove (2.30) ≅ (2.31) directly.

minor comments (3)

[§4.3.13] 'By Remark 4.3.12' should presumably read 'By Remark 4.1.9'; there is no Remark 4.3.12 in the manuscript.
[Introduction, Theorem A and Theorem B] The displayed statements of Theorem A and Theorem B contain corrupted arrow glyphs such as '≃ /leftr⫯g⊸tl⫯ne →'; these must be fixed in the final version.
[§4.4.2] In the sentence following (4.30), the direction and induced action of the map T~^♯ → T~_β should be spelled out more explicitly; currently the reader must infer the action from Lemma 4.3.6.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem B(2) is a genuine derived identity, but its derivation chain leans pervasively on the second author's prior preprint [Zha22], and §2.3.6 contains an omitted independence verification.

self citation load bearing [§1.2.2, eq. (1.6); used throughout §§2–4 (e.g., Prop 1.3.6, Prop 3.1.3, Thm 4.3.9)]
"The construction of [Zha22, §2.1] yields a Z-linear functor ∫F ∶ Maps_e(BG,B^4A(1)) → Cov(G(F),A).(1.6)"

The paper's central objects—the cover ̃G_β defined via (1.8), the L-group ̃H, and the torus duality (2.6) used to define the central-core character map (2.40) and hence Ω_β—are all built on the functor (1.6) and on results imported from [Zha22] (e.g., Props 4.4.5, 4.6.2, 4.6.6, 4.7.3, 5.5.4; Thm 3.1.5). [Zha22] is a preprint by the second author and is not reproved or machine-checked here. The derivation of Theorem B(2) therefore terminates in a load-bearing self-citation: if the imported framework were wrong, Ω_β and γ would not be well-defined. This is a circularity burden, although the final identity (4.28) is a new computation and not literally identical to any single cited statement.

full rationale

The central result, Corollary 4.3.12 / Theorem B(2), is a derived formula: Ω_β(σ)/Ω(σ) is computed in Theorem 4.3.9 as γ(Kott(β)) using the canonical quadratic structure (Proposition 3.1.3) and the Kottwitz–Tate compatibility (4.26). This is not a fitted parameter renamed as a prediction, and no empirical input is involved. The assumptions about LLC_β are explicitly conjectural (diagram (4.27), Lemma 4.3.11), so the paper does not disguise conjecture as theorem. The main circularity burden is foundational: the ∫_F functor (1.6), the classification of étale metaplectic covers (1.19), the sharp-cover construction (1.20)–(1.22), and several technical propositions used in §2–§3 are imported from [Zha22], a preprint by the second author, without reproof. This self-citation is load-bearing, but the final formula still requires new work (e.g., the canonical quadratic structure and the comparison (4.24)), so the central claim has independent content beyond the cited framework. Separately, the omitted independence verification in §2.3.6 is a genuine correctness gap: (2.28) must be independent of F1 and µ′ for (2.7), (2.40), and Ω_β to be well-defined. This is a gap rather than a circular reduction, but it further weakens the self-containedness of the derivation chain. Overall score 4.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper is pure mathematics and introduces no fitted parameters. The axioms are the standard cohomological and ∞-categorical background, the imported étale metaplectic formalism of [Zha22] (prior work of the second author), Kottwitz's isocrystal theory, and the conjectural local Langlands compatibility needed to connect the obstruction to actual L-packets. The invented entities are mathematical constructions, not empirical postulates, so none has independent evidence outside the paper.

assumptions (5)

domain assumption The foundational functor ∫_F (1.6) of [Zha22, §2.1] exists and has the stated Z-linear properties.
All covers G~_β are defined by this functor; the paper does not reprove its construction.
standard math Kottwitz's isocrystal theory, including the bijection (1.5) between basic G-isocrystals and Galois coinvariants, is valid.
Used to define extended pure inner forms and to translate the vanishing criterion into the torsor statement in Theorem B.
domain assumption The conjectural local Langlands correspondence for covers (Conjecture 1.4.14) and its inner-form version (Conjecture 1.4.16), together with compatibility with central core characters (diagrams (4.6), (4.27)), are assumed when interpreting Ω_β as governing nonempty L-packets.
The paper explicitly labels these as conjectural; the formal theorems do not require them, but the advertised explanation of empty L-packets does.
standard math The higher algebra of Lurie (∞-categories, E_∞-monoids, connective spectra) is taken as background.
The entire formulation uses this language; see §0.3.
standard math Tate duality, Artin reciprocity, the Hilbert symbol, and the vanishing of H^2(W_F, ˇT(C)) and H^3(W_F, T_H,sc(C)) are valid.
Used in the proof of Theorem 2.1.6, e.g., §2.2.5 and §2.4.4, citing [Kar13] and [Ser79].

invented entities (2)

The cover G~_β of the extended pure inner form G_β(F)
purpose: To provide a home for the L-parameter σ when the trivial form's obstruction is nonvanishing, thereby 'finding' missing L-packets.
Constructed canonically from (1.8); no empirical handle outside the paper.
Weissman's obstruction Ω_β(σ)
purpose: A character of K encoding whether the central core characters of G~ and G~_β are compatible for σ; its vanishing is the criterion in Theorem B.
Defined in §4.3.10; predictive power depends on the conjectural compatibility diagrams (4.6)/(4.27).

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Cite this review

Pith. "Pith review of What are the extended pure inner forms of a cover?." pith.science (2026). https://pith.science/paper/P2L6TRDG

@misc{pith2026250608696,
  author       = {Pith},
  title        = {Pith review of: What are the extended pure inner forms of a cover?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2L6TRDG}},
  note         = {Machine review of arXiv:2506.08696}
}

read the original abstract

Kottwitz suggested to study all extended pure inner forms together in the local Langlands correspondence for linear reductive groups. We extend this philosophy to a large class of covers, including those defined by Brylinski and Deligne, and explain its relation with Weissman's observation that L-packets for covers are sometimes empty.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 19 canonical work pages

[1]

Jean-Luc Brylinski and Pierre Deligne, Central extensions of reductive groups by K_2 , Publ. Math. Inst. Hautes \' E tudes Sci. (2001), no. 94, 5--85. 1896177

work page 2001
[2]

Borel and T

A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II , Tohoku Math. J. (2) 20 (1968), 443--497. 244259

work page 1968
[3]

Hautes \' E tudes Sci

Pierre Deligne, Extensions centrales de groupes alg\' e briques simplement connexes et cohomologie galoisienne , Inst. Hautes \' E tudes Sci. Publ. Math. (1996), no. 84, 35--89 (1997). 1441006

work page 1996
[4]

398, Soci\' e t\' e Math\' e matique de France, Paris, 2018, L-groups and the Langlands program for covering groups, pp

Wee Teck Gan and Fan Gao, The L anglands- W eissman program for B rylinski- D eligne extensions , L-groups and the L anglands program for covering groups, no. 398, Soci\' e t\' e Math\' e matique de France, Paris, 2018, L-groups and the Langlands program for covering groups, pp. 187--275. 3802419

work page 2018
[5]

Weissman, L-groups and the L anglands program for covering groups: a historical introduction , L-groups and the L anglands program for covering groups, no

Wee Teck Gan, Fan Gao, and Martin H. Weissman, L-groups and the L anglands program for covering groups: a historical introduction , L-groups and the L anglands program for covering groups, no. 398, Soci\' e t\' e Math\' e matique de France, Paris, 2018, L-groups and the Langlands program for covering groups, pp. 1--31. 3802417

work page 2018
[6]

(N.S.) 24 (2018), no

Dennis Gaitsgory and Sergey Lysenko, Parameters and duality for the metaplectic geometric L anglands theory , Selecta Math. (N.S.) 24 (2018), no. 1, 227--301, References are to the corrected version, available at: https://lysenko.perso.math.cnrs.fr. 3769731

work page 2018
[7]

Tasho Kaletha, Supercuspidal L -packets via isocrystals , Amer. J. Math. 136 (2014), no. 1, 203--239. 3163358

work page 2014
[8]

Tasho Kaletha, Covers of reductive groups and functoriality, https://arxiv.org/abs/2209.14357, 2022, arXiv preprint

arXiv 2022

Show all 22 references

[9]

Karpuk, Cohomology of the W eil group of a p -adic field , J

David A. Karpuk, Cohomology of the W eil group of a p -adic field , J. Number Theory 133 (2013), no. 4, 1270--1288. 3003999

2013
[10]

Kottwitz, Isocrystals with additional structure, Compositio Math

Robert E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201--220. 809866

1985
[11]

II , Compositio Math

, Isocrystals with additional structure. II , Compositio Math. 109 (1997), no. 3, 255--339. 1485921

1997
[12]

170, Princeton University Press, Princeton, NJ, 2009

Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. 2522659

2009
[13]

, Higher algebra, https://people.math.harvard.edu/ lurie/papers/HA.pdf, 2017, preprint

2017
[14]

139, Academic Press, Inc., Boston, MA, 1994, Translated from the 1991 Russian original by Rachel Rowen

Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994, Translated from the 1991 Russian original by Rachel Rowen. 1278263

1994
[15]

67, Springer-Verlag, New York-Berlin, 1979, Translated from the French by Marvin Jay Greenberg

Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979, Translated from the French by Marvin Jay Greenberg. 554237

1979
[16]

The Stacks Project Authors , Stacks Project , https://stacks.math.columbia.edu, 2018

2018
[17]

Hautes \' E tudes Sci

Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes \' E tudes Sci. Publ. Math. (1965), no. 25, 49--80. 180554

1965
[18]

Weissman, Metaplectic tori over local fields, Pacific J

Martin H. Weissman, Metaplectic tori over local fields, Pacific J. Math. 241 (2009), no. 1, 169--200. 2485462

2009
[19]

, Covers of tori over local and global fields, Amer. J. Math. 138 (2016), no. 6, 1533--1573. 3595494

2016
[20]

398, S oci \'e t \'e M ath \'e matique de F rance, 2018, L-groups and the Langlands program for covering groups, pp

, L -groups and parameters for covering groups , Ast\' e risque, no. 398, S oci \'e t \'e M ath \'e matique de F rance, 2018, L-groups and the Langlands program for covering groups, pp. 33--186. 3802418

2018
[21]

Monogr., vol

Jiu-Kang Yu, On the local L anglands correspondence for tori , Ottawa lectures on admissible representations of reductive p -adic groups, Fields Inst. Monogr., vol. 26, Amer. Math. Soc., Providence, RI, 2009, pp. 177--183. 2508725

2009
[22]

Yifei Zhao, \' E tale metaplectic covers of reductive group schemes , https://arxiv.org/abs/2204.00610, 2022

2022 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Jean-Luc Brylinski and Pierre Deligne, Central extensions of reductive groups by K_2 , Publ. Math. Inst. Hautes \' E tudes Sci. (2001), no. 94, 5--85. 1896177

work page 2001

[2] [2]

Borel and T

A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II , Tohoku Math. J. (2) 20 (1968), 443--497. 244259

work page 1968

[3] [3]

Hautes \' E tudes Sci

Pierre Deligne, Extensions centrales de groupes alg\' e briques simplement connexes et cohomologie galoisienne , Inst. Hautes \' E tudes Sci. Publ. Math. (1996), no. 84, 35--89 (1997). 1441006

work page 1996

[4] [4]

398, Soci\' e t\' e Math\' e matique de France, Paris, 2018, L-groups and the Langlands program for covering groups, pp

Wee Teck Gan and Fan Gao, The L anglands- W eissman program for B rylinski- D eligne extensions , L-groups and the L anglands program for covering groups, no. 398, Soci\' e t\' e Math\' e matique de France, Paris, 2018, L-groups and the Langlands program for covering groups, pp. 187--275. 3802419

work page 2018

[5] [5]

Weissman, L-groups and the L anglands program for covering groups: a historical introduction , L-groups and the L anglands program for covering groups, no

Wee Teck Gan, Fan Gao, and Martin H. Weissman, L-groups and the L anglands program for covering groups: a historical introduction , L-groups and the L anglands program for covering groups, no. 398, Soci\' e t\' e Math\' e matique de France, Paris, 2018, L-groups and the Langlands program for covering groups, pp. 1--31. 3802417

work page 2018

[6] [6]

(N.S.) 24 (2018), no

Dennis Gaitsgory and Sergey Lysenko, Parameters and duality for the metaplectic geometric L anglands theory , Selecta Math. (N.S.) 24 (2018), no. 1, 227--301, References are to the corrected version, available at: https://lysenko.perso.math.cnrs.fr. 3769731

work page 2018

[7] [7]

Tasho Kaletha, Supercuspidal L -packets via isocrystals , Amer. J. Math. 136 (2014), no. 1, 203--239. 3163358

work page 2014

[8] [8]

Tasho Kaletha, Covers of reductive groups and functoriality, https://arxiv.org/abs/2209.14357, 2022, arXiv preprint

arXiv 2022