Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Green functions for positive-depth Deligne--Lusztig induction

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For sufficiently large residue fields, positive-depth Deligne–Lusztig induction is the geometric realization of the Howe-unramified regular supercuspidal L-packets, and its Green functions extend the match to all characters.

desk verdict A serious comparison theorem for positive-depth Deligne–Lusztig induction, conditional on Chan's scalar-product preprint; it deserves refereeing, but the referee must verify that dependency. read the letter →

arxiv 2506.04449 v1 pith:RAE7JMIQ submitted 2025-06-04 math.RT math.NT

Charlotte Chan , Masao Oi This is my paper
classification math.RTmath.NT MSC 22E5020G4011F70
keywords p-adicreductivegroupspositive-depthDeligne–LusztiginductionGreenfunctionssupercuspidalrepresentationsL-packetsHowefactorizationparahoricsubgroupslocalLanglandscorrespondence
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

This paper aims to show that positive-depth Deligne–Lusztig induction is not merely a formal analogue of the algebraic construction of supercuspidal representations but a genuine geometric realization of it. For residue fields large enough, the compact induction of the geometrically defined virtual representation $R^{G_r}_{T_r}(\theta)$ is exactly the irreducible supercuspidal representation obtained from the algebraic construction, with the sign twist $\varepsilon_{\mathrm{ram}}$ appearing automatically rather than inserted by hand. The paper then extends the match from characters with trivial Weyl stabilizer to arbitrary characters by proving that the Green functions of geometric and algebraic origin coincide. If the results are correct, one geometric machine produces the correct supercuspidal L-packets and simultaneously explains, through a positive-depth Springer hypothesis, why orbital integrals appear in supercuspidal character formulas.

What carries the argument

The machinery consists of three objects working together. First is the positive-depth Deligne–Lusztig induction functor $R^{G_r}_{T_r}(\theta)$, defined by the $\theta$-isotypic cohomology of the variety $X_{T_r\subset G_r}$; its character at an unramified very regular element is the sum $\sum_{w\in W_{G_{x,0}}(T_\gamma,T)} \theta^w(\gamma)$. Second is the “litmus test” uniqueness theorem: under the largeness inequality $(*)$, at most one irreducible parahoric representation can have that character shape, a fact proved by bounding the non-very-regular contribution through Cauchy–Schwarz. Third are the positive-depth Green functions $Q^{G_r}_{T_r}(\theta_+)$—character values at unipotent elements—whose orthogonality and comparison formulas carry the result from regular $\theta$ to all $\theta$ and, via Fourier transform of coadjoint-orbit delta functions, to the Springer hypothesis.

What would settle it

Check the scalar-product formula by computing $\langle R^{G_r}_{T_r}(\theta), R^{G_r}_{T'_r}(\theta')\rangle$ for an unramified elliptic Howe-factorizable pair and comparing it with the number of Weyl-group elements sending $\theta$ to $\theta'$; any mismatch invalidates the irreducibility step. Alternatively, in the excluded small-$q$ cases $G_2$ over $\mathbb{F}_3$ or $\mathbb{F}_5$ with the Coxeter torus, search for a non-unipotent irreducible representation whose values on all regular semisimple elements equal $\pm\Theta_{R^G_T(\theta)}$ without being isomorphic to $\pm R^G_T(\theta)$, which would refute the characterization theorem outside the largeness range.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the isomorphism of $T G_{x,0}$-representations $\tau^{\mathrm{FKS}}_{\Psi} \cong (-1)^{r(G_0)-r(T)+r(T,\theta)} R^{G_r}_{T_r}(\theta)$ for every regular $\theta$ attached to an unramified elliptic pair, under odd non-bad $p$, the stated divisibility conditions, and the largeness inequality $(*)$. Because the right-hand side is the geometric object and the left-hand side is the twisted algebraic object, compact induction yields $\pi^{\mathrm{FKS}}_{\Psi}$ as the irreducible supercuspidal representation corresponding to the pair $(T,\theta\cdot\varepsilon_{\mathrm{ram}})$, so positive-depth Deligne–Lusztig induction realizes the Howe-unramified regular supercuspidal L-packet. The comparison for arbitrary $\theta$ follows by matching the unipotent restrictions of the two families, giving $\circ\tau^{\mathrm{FKS}}_{(T,\theta)} \cong (-1)^{r(G_0)-r(T)+r(T,\theta)} R^{G_r}_{T_r}(\theta)$ for all unramified elliptic pairs.

Load-bearing premise

The load-bearing premise is the quoted scalar-product/Mackey formula for positive-depth Deligne–Lusztig induction, taken from another preprint and not reproved in this paper; if that formula fails in an unexamined case, the irreducibility step on which the main comparison theorems rest collapses.

Editorial extensions

If this is right

  • For regular $\theta$ and large $q$, $\mathrm{c-Ind}\big(R^{G_r}_{T_r}(\theta)\big)$ is the irreducible supercuspidal representation $\pi^{\mathrm{FKS}}_{\Psi}$, so the geometric functor realizes the Howe-unramified regular supercuspidal L-packets; in particular the assignment is compatible with the local Langlands correspondence.
  • For arbitrary $\theta$, the algebraic virtual representation $\circ\tau^{\mathrm{FKS}}_{(T,\theta)}$ and the geometric representation $(-1)^{r(G_0)-r(T)+r(T,\theta)}R^{G_r}_{T_r}(\theta)$ are isomorphic, so the regular case determines the whole Howe-unramified family.
  • Every Howe-unramified supercuspidal type appears in some $R^{G_r}_{T_r}(\theta)$, and Howe-unramified Kim–Yu types occur in the cohomology of the positive-depth Deligne–Lusztig varieties.
  • For 0-toral Howe-unramified regular pairs, the positive-depth Green function is the Fourier transform of the delta function on a coadjoint orbit, giving a geometric proof of the orbital-integral character formula.
  • The same comparison shows that positive-depth Deligne–Lusztig induction preserves stability: it maps stable conjugacy classes of unramified elliptic regular pairs to stable distributions.

Reading between the lines

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

  • Because the uniqueness theorem needs no Howe factorization and no restriction on $p$, the geometric side of the comparison is available in settings where the algebraic construction does not yet exist; the first testable payoff would be a comparison for non-Howe-factorizable characters or for $p=2$ once such representations are constructed.
  • The paper's small-$q$ analysis shows that failures of the characterization in $G_2$ over $\mathbb{F}_3$ and $\mathbb{F}_5$ all come from unipotent representations; this suggests a general principle, made precise for depth zero in the paper, that non-unipotent representations are pinned by regular-semisimple character values even when the largeness inequality fails.
  • The orthogonality relations for positive-depth Green functions are a reusable structure: they should feed directly into endoscopic character identities for positive-depth supercuspidal representations, in the same way classical Green functions do for depth zero.
  • The 0-toral Springer hypothesis is a proof of concept; once the trace-of-Frobenius computation for character sheaves is available for non-0-toral $\theta$, the same argument should prove the full positive-depth Springer hypothesis and rederive the general supercuspidal character formula geometrically.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper develops a positive-depth analogue of Deligne--Lusztig induction for unramified elliptic pairs (T, θ) in a p-adic reductive group, under a largeness assumption on the residue field size q. The central result is Theorem 5.6: for regular θ and q satisfying Henniart's inequality (∗), the Fintzen--Kaletha--Spice twisted Yu representation τ^{FKS}_Ψ is isomorphic to (−1)^{r(G0)−r(T)+r(T,θ)} R^G_r_{T_r}(θ), so that compact induction of the latter is the expected supercuspidal representation. This is obtained from a ``litmus test'' characterization theorem (Theorem 3.2) proved by a Cauchy--Schwarz estimate, together with character formulas for the algebraic side (Proposition 4.3) and for positive-depth Deligne--Lusztig induction (Proposition 5.5). The paper then defines Green functions for both the geometric and algebraic constructions, proves an orthogonality relation (Proposition 6.9), and uses it to extend the comparison from regular θ to arbitrary θ (Theorem 8.3, Corollary 8.4). A further application is a positive-depth Springer hypothesis in the 0-toral setting (Theorem 9.3) and a geometric derivation of a supercuspidal character formula. The final section presents a detailed small-q analysis for G2, showing that failures of the characterization are intimately tied to unipotent representations.

Significance. If the central comparison is correct, the paper gives a geometric realization of Kaletha's Howe-unramified regular supercuspidal L-packets, with the ε_ram twist appearing automatically rather than as an external correction. The characterization theorem (Theorem 3.2) is a new and potentially widely applicable tool, and the paper explicitly demonstrates its sharpness through the G2 small-q analysis. The Green-function orthogonality (Proposition 6.9) and the character formula for FKS--Yu virtual representations (Theorem 7.2) are independent structural contributions. The paper also contains concrete machine-verifiable character table data for G2(F3) and G2(F5), which is a commendable feature. However, the main comparison and the Section 9 applications rest on unrefereed preprint results ([Cha24, Theorem 6.2] and [BC24, Theorem 10.9]), and one central theorem (Theorem 8.2) is stated with a notational collision that makes it formally tautological. These issues are load-bearing rather than cosmetic, so the current version needs revision.

major comments (4)
  1. [Section 5.1, Theorem 5.2] The proof of Theorem 5.6 begins with 'By Theorem 5.2, either R^G_r_T_r(θ) or −R^G_r_T_r(θ) is an irreducible representation of G_{x,0}', and Theorem 5.2 is quoted verbatim from [Cha24, Theorem 6.2], a preprint by the first author that is not reproved in this paper. This is load-bearing: it is the only input guaranteeing that ±R^G_r_T_r(θ) is irreducible, which Theorem 3.2 then needs to identify with the FKS construction. The same theorem is also used in the proof of Proposition 6.9 and hence in Theorem 8.2, Theorem 8.3, and Corollary 8.4. If [Cha24, Theorem 6.2] carries an unstated hypothesis or is not yet available in refereed form, the main comparison of the paper is unsupported. Please include a proof of the needed case (or of a statement sufficient for unramified elliptic Howe-factorizable pairs) or explicitly reformulate the main theorems as conditional on [Cha24, Theorem 6.2] and state its current status.
  2. [Section 8, Theorem 8.2] The statement 'QGr_Tr(θ+) = (−1)^{r(G0)−r(T)+r(T,θ)} · QGr_Tr(θ+)' is formally a tautology because the symbol QGr_Tr is used for the geometric Green function (Definition 6.4) and the FKS–Yu Green function (Definition 7.1) without distinction. As written, the theorem asserts X = c·X, which can only hold when c = 1, but the sign is generally nontrivial. The proof makes it clear that the intended assertion is an equality between the two different Green functions. Please introduce distinct notations, for example Q^{geom} and Q^{FKS}, and restate Theorem 8.2, Theorem 8.3, and Corollary 8.4 accordingly.
  3. [Section 8, proof of Theorem 8.2] The proof asserts that θ′ := θ·φ^{-1}_{-1}·φ′_{-1} is a regular character whenever φ′_{-1} is a regular depth-zero character of T. This is not automatic: for a nontrivial w ∈ W_G(T), the condition (θ·χ)^w = θ·χ is equivalent to χ^w χ^{-1} = θ^w θ^{-1}, so the set of χ making θ·χ non-regular is a finite union of cosets of proper subgroups of the character group. The existence of a suitable φ′_{-1} requires an argument using the largeness of q beyond the assumptions already stated in Section 2.1. Please add a short counting argument or a reference.
  4. [Section 9, Theorem 9.3] The proof of the positive-depth Springer hypothesis uses [BC24, Theorem 10.9] to identify the function associated with pInd^{G_r}_{T_r}(L_θ) with the character of R^G_r_T_r(θ). This is a second load-bearing dependency on an unpublished preprint. Since the results of Section 9 — in particular Corollary 9.9 — depend on this identification, please either prove the needed statement, provide a precise reference to a publicly available version with theorem numbers, or clearly mark the results of Section 9 as conditional on [BC24].
minor comments (5)
  1. [Section 5.3, Theorem 5.8] The notation |R^G_{j,r}_{T_{j,r}}(jθ)| is used to denote the sign-adjusted irreducible component of a virtual representation; this notation is only defined implicitly in the proof of Theorem 5.6. Please define it before first use in Theorem 5.8.
  2. [Section 10.1.4] There is a numerical typo: the dimension of R^G_T(θ) for G2(F5) is computed as (q−1)^2(q+1)^2(q^2+q+1) = 17856, but the text then refers to 'three irreducible representations whose dimension is 17586'. Please correct the typo and verify the character labels.
  3. [Section 10, Table 4] Table 4 is difficult to read because the rows and columns are not visually aligned; the values for different conjugacy classes run together. Consider formatting the character table as a proper matrix with clear column separations.
  4. [Section 5.5, Conjecture 5.12] The notation |R^G_r_{G'_r}(α ⊗ φ)| in Conjecture 5.12 is used before being defined; please clarify that it denotes the unique irreducible component up to sign of the virtual representation.
  5. [Section 3, Theorem 3.2] The phrase 'for some sign constant c ∈ {±1}' should be clarified: c is allowed to depend on the representation π, and the conclusion of the theorem forces the two constants to coincide. Stating this explicitly would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the isomorphism is between independently defined geometric and algebraic objects, with the uniqueness theorem proved in the paper; cited same-author results are external dependencies, not reductions.

full rationale

The derivation chain is not circular. Theorem 5.6 compares the cohomological functor R^G_r_{T_r}(θ) (defined in Section 5.1) with the algebraic FKS/Yu representation ◦τ^FKS_Ψ (Section 4.2). The bridge is the new 'litmus test' Theorem 3.2, proved in this paper using Cauchy–Schwarz and the largeness condition (∗); no parameter is fitted to force the equality, and the character values on very regular elements for the two sides are supplied by separate statements (Propositions 5.5 and 4.3) rather than by the isomorphism being proved. The sign (−1)^{r(G0)−r(T)+r(T,θ)} is not chosen to match: the FKS side already contains the twist ϵΨ by construction, and the isomorphism determines the sign. The Green-function results (Theorems 8.2, 8.3, Corollary 8.4) are consequences of the regular case, not assumptions. The manuscript does contain load-bearing dependencies on same-author work: Theorem 5.2 is quoted verbatim from [Cha24, Theorem 6.2] ('By Theorem 5.2, either R^G_r_{T_r}(θ) or −R^G_r_{T_r}(θ) is an irreducible representation of G_{x,0}'), and the paper states it was 'awaiting the resolution of the results in [Cha24]' (Section 1); Section 9 similarly relies on [BC24, Theorem 10.9] for the trace-of-Frobenius computation, with the paper noting the constraint is 'at present only written down for 0-toral θ'. These are external mathematical inputs about scalar products and character sheaves, not reductions of the target isomorphism to itself, so they are verification gaps rather than circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central results rest on four kinds of input. First, explicit hypotheses on p and q, all stated in Section 2.1: odd non-bad p with the divisibility conditions, the Henniart inequality (*), and the existence of regular semisimple elements and regular characters in T0(Fq). Second, classical theorems quoted from the literature: Deligne-Lusztig theory, the parahoric character formula [CI21], Yu's construction [Yu01], Kaletha's L-packets [Kal19], and the FKS twist [FKS23]. Third, recent results from the authors' own program: [Cha24] for the scalar product formula, [BC24] for character sheaf trace formulas, and [CO23] plus [CO25] for the 0-toral comparison and the very-regular character formulas. Fourth, finite group computations for G2 over F3 and F5, sourced from GAP3. No parameter is fitted to data; the q-bound is a sufficient condition, not a tuned constant. No new entity is postulated; the positive-depth Green functions and the FKS-Yu virtual representations are defined constructs, not unexplained inputs.

assumptions (8)
  • domain assumption Henniart inequality (*): |T0(Fq)| / |T0(Fq)_nvreg| > 2 * |W_{G0}(T0)(Fq)|, i.e., q sufficiently large.
    Standing largeness assumption for Theorem 3.2 and all comparison results, Section 2.1(3). The paper proves sufficiency and shows in Section 10 that the bound is near necessary, with G2 small-q counterexamples.
  • domain assumption p is odd and not bad for G, and p does not divide |pi1(G_der)| * |pi1(Ghat_der)|.
    Section 2.1(1). Guarantees Howe factorization for every tame pair and validity of Kaletha's regular supercuspidal construction [Kal19]. The authors note this is convenient, not necessary.
  • domain assumption F has characteristic 0 for the Langlands discussion, and p >= (2+e)n so that log and exp maps exist.
    Sections 5.3 and 9.1.1. Needed for the L-packet parametrization and for the p-adic harmonic analysis in the Springer hypothesis section.
  • domain assumption For large q, T0(Fq) contains a regular semisimple element and a regular character.
    Sections 2.1(4)-(5) and Remark 8.1. Weaker than (*); required for Theorem 9.3 and for the choice of a regular depth-zero character in Theorem 8.2.
  • standard math [Cha24, Theorem 6.2] scalar product and Mackey formula for positive-depth Deligne-Lusztig induction.
    Quoted as Theorem 5.2. Implies R^G_r_{T_r}(theta) is irreducible up to sign for elliptic Howe-factorizable pairs; a preprint result not reproved here. Carries the existence side of the comparison.
  • standard math [CI21, Theorem 1.2] character formula of R^G_r_{T_r}(theta) on unramified very regular elements.
    Quoted as Proposition 5.5. Gives the geometric side's character values used in the application of Theorem 3.2.
  • standard math [CO25, Proposition 4.11] and [FKS23] character formula for Yu and FKS supercuspidals on very regular elements.
    Quoted as Propositions 4.2 and 4.3. Gives the algebraic side's character values; the epsilon_ram twist is absorbed in the FKS variant.
  • standard math [BC24, Theorem 10.9] trace of Frobenius of positive-depth character sheaves.
    Used in the proof of Theorem 9.3. The paper states it is currently written only for 0-toral theta, exactly the setting of Section 9.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Green functions for positive-depth Deligne--Lusztig induction." pith.science (2026). https://pith.science/paper/RAE7JMIQ

@misc{pith2026250604449,
  author       = {Pith},
  title        = {Pith review of: Green functions for positive-depth Deligne--Lusztig induction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RAE7JMIQ}},
  note         = {Machine review of arXiv:2506.04449}
}
abstract

Under a largeness assumption on the size of the residue field, we give an explicit description of the positive-depth Deligne--Lusztig induction of unramified elliptic pairs $(T,\theta)$. When $\theta$ is regular, we show that positive-depth Deligne--Lusztig induction gives a geometric realization of Kaletha's Howe-unramified regular $L$-packets. This is obtained as an immediate corollary of a very simple "litmus test" characterization theorem which we foresee will have interesting future applications to small-$p$ constructions. We next define and analyze Green functions of two different origins: Yu's construction (algebra) and positive-depth Deligne--Lusztig induction (geometry). Using this, we deduce a comparison result for arbitrary $\theta$ from the regular setting. As a further application of our comparison isomorphism, we prove the positive-depth Springer hypothesis in the $0$-toral setting and use it to give a geometric explanation for the appearance of orbital integrals in supercuspidal character formulae.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An explicit decomposition of higher Deligne-Lsuztig representations

    math.RT 2025-06 conditional novelty 6.0 of 10

    For q ≥ c_Λ with 2 ≤ c_Λ ≤ 4, the geometric representation κ_Λ from cohomology of higher Deligne-Lusztig varieties equals κ(Λ)⊗ϵ_Λ, giving an explicit irreducible decomposition of elliptic higher Deligne-Lusztig repre...

Reference graph

Works this paper leans on

22 extracted references · 16 canonical work pages · cited by 1 Pith paper

  1. [1]

    [AD04] J. D. Adler and S. DeBacker, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups , J. Reine Angew. Math. 575 (2004), 1–35. [Adl98] J. D. Adler, Refined anisotropic K-types and supercuspidal representations , Pacific J. Math. 185 (1998), no. 1, 1–32. [AFMO24a] J. D. Adler, J. Fintzen, M. Mishra, and K. Ohara, Re...

    arXiv 2004

  2. [3]

    [AS08] J. D. Adler and L. Spice, Good product expansions for tame elements of p-adic groups , Int. Math. Res. Pap. IMRP (2008), no. 1, Art. ID rp. 003,

    work page 2008

  3. [5]

    [BH96] C. J. Bushnell and G. Henniart, Local tame lifting for GL(N ). I. Simple characters , Inst. Hautes ´Etudes Sci. Publ. Math. (1996), no. 83, 105–233. [BK98] C. J. Bushnell and P. C. Kutzko, Smooth representations of reductive p-adic groups: structure theory via types , Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. [Car72] R. W. Carter, Conj...

    arXiv 1996

  4. [6]

    Characterization of supercuspidal representations and very regular elements

    [CI21] C. Chan and A. Ivanov, Cohomological representations of parahoric subgroups , Represent. Theory 25 (2021), 1–26. [CO23] C. Chan and M. Oi, Characterization of supercuspidal representations and very regular ele- ments, preprint, arXiv:2301.09812,

    work page Pith review arXiv 2021

  5. [8]

    Dudas and A

    [DI24] O. Dudas and A. B. Ivanov, Orthogonality relations for deep level Deligne-Lusztig schemes of Coxeter type, Forum Math. Sigma 12 (2024), Paper No. e66,

    work page 2024

  6. [13]

    [INT24] A. B. Ivanov, S. Nie, and P. Tan, Deep level Deligne-Lusztig representations of Coxeter type , preprint, arXiv:2407.18694v2,

    work page Pith review arXiv

  7. [14]

    Kaletha, Endoscopic character identities for depth-zero supercuspidal L-packets, Duke Math

    [Kal11] T. Kaletha, Endoscopic character identities for depth-zero supercuspidal L-packets, Duke Math. J. 158 (2011), no. 2, 161–224. [Kal19] , Regular supercuspidal representations, J. Amer. Math. Soc. 32 (2019), no. 4, 1071–

    work page 2011

  8. [16]

    Kazhdan and Y

    [KV06] D. Kazhdan and Y. Varshavsky, Endoscopic decomposition of certain depth zero representa- tions, Studies in Lie theory, Progr. Math., vol. 243, Birkh¨ auser Boston, Boston, MA, 2006, pp. 223–301. [KW01] R. Kiehl and R. Weissauer, Weil conjectures, perverse sheaves and l’adic Fourier transform , Ergebnisse der Mathematik und ihrer Grenzgebiete

    work page 2006

Show all 22 references
  1. [27]

    Deligne and G

    55 [DL76] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. [DR09] S. DeBacker and M. Reeder, Depth-zero supercuspidal L-packets and their stability , Ann. of Math. (2) 169 (2009), no. 3, 795–901. [...

    1976 arXiv

  2. [95]

    [AS09] , Supercuspidal characters of reductive p-adic groups, Amer. J. Math. 131 (2009), no. 4, 1137–1210. [BC24] R. Bezrukavnikov and C. Chan, Generic character sheaves on parahoric subgroups , preprint, arXiv:2401.07189,

    2009 arXiv

  3. [166]

    [IN25] A. B. Ivanov and S. Nie, Convex elements and deep level Deligne-Lusztig varieties , preprint, arXiv:2503.13412v1,

    arXiv

  4. [981]

    Moy and G

    [MP96] A. Moy and G. Prasad, Jacquet functors and unrefined minimal K-types, Comment. Math. Helv. 71 (1996), no. 1, 98–121. 56 [Nie24] S. Nie, Decomposition of higher Deligne-Lusztig representations , preprint, arXiv:2406. 06430v1,

    1996

  5. [1170]

    Kazhdan, Proof of Springer’s hypothesis , Israel J

    [Kaz77] D. Kazhdan, Proof of Springer’s hypothesis , Israel J. Math. 28 (1977), no. 4, 272–286. [KP23] T. Kaletha and G. Prasad, Bruhat-Tits theory—a new approach , New Mathematical Mono- graphs, vol. 44, Cambridge University Press, Cambridge,

    1977

  6. [1968]

    Yu, Construction of tame supercuspidal representations , J

    [Yu01] J.-K. Yu, Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), no. 3, 579–622. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48105, USA. Email address : charchan@umich.edu Department of...

    2001

  7. [1984]

    [Lus90] , Green functions and character sheaves, Ann. of Math. (2) 131 (1990), no. 2, 355–408. [Lus04] , Representations of reductive groups over finite rings , Represent. Theory 8 (2004), 1–14. [Mir04] I. Mirkovi´ c, Character sheaves on reductive Lie algebras , Mosc. Math. J...

    1990

  8. [1997]

    Spice, Topological Jordan decompositions, J

    [Spi08] L. Spice, Topological Jordan decompositions, J. Algebra 319 (2008), no. 8, 3141–3163. [Spi18] , Explicit asymptotic expansions for tame supercuspidal characters, Compos. Math. 154 (2018), no. 11, 2305–2378. [Spi21] , Explicit asymptotic expansions in p-adic harmonic an...

    2008

  9. [2001]

    Kim and J.-K

    [KY17] J.-L. Kim and J.-K. Yu, Construction of tame types , Representation theory, number theory, and invariant theory, Progr. Math., vol. 323, Birkh¨ auser/Springer, Cham, 2017, pp. 337–357. [L¨ ub19] F. L¨ ubeck, Centralizers and numbers of semisimple classes in exceptional ...

    2017

  10. [2020]

    162, Springer-Verlag, Berlin-New York, 1970, Notes by G

    [HC70] Harish-Chandra, Harmonic analysis on reductive p-adic groups, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin-New York, 1970, Notes by G. van Dijk. [Hen92] G. Henniart, Correspondance de Langlands-Kazhdan explicite dans le cas non ramifi´ e, Math. Nachr....

    1992

  11. [2021]

    Stasinski, Unramified representations of reductive groups over finite rings , Represent

    [Sta09] A. Stasinski, Unramified representations of reductive groups over finite rings , Represent. The- ory 13 (2009), 636–656. [Ste68] R. Steinberg, Endomorphisms of linear algebraic groups , Memoirs of the American Mathemat- ical Society, No. 80, American Mathematical Socie...

    2009

  12. [2023]

    [CO25] , Geometric L-packets of Howe-unramified toral supercuspidal representations , J. Eur. Math. Soc. (JEMS) 27 (2025), no. 1, 1465–1526. [CS17] Z. Chen and A. Stasinski, The algebraisation of higher Deligne-Lusztig representations, Selecta Math. (N.S.) 23 (2017), no. 4, 29...

    2025 arXiv

  13. [2024]

    [AFMO24b] , Structure of Hecke algebras arising from types , preprint, arXiv:2408.07801v1,

    arXiv

  14. [2025]

    G´ erardin, Construction de s´ eries discr` etesp-adiques, Lecture Notes in Mathematics, Vol

    [G´ er75] P. G´ erardin, Construction de s´ eries discr` etesp-adiques, Lecture Notes in Mathematics, Vol. 462, Springer-Verlag, Berlin-New York, 1975, Sur les s´ eries discr` etes non ramifi´ ees des groupes r´ eductifs d´ eploy´ esp-adiques. [G´ er77] , Weil representations ...

    1977

Pith tools

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