Construction of tame supercuspidal representations in arbitrary residue characteristic

David Schwein; Jessica Fintzen

REVIEW 2 major objections 3 minor 1 cited by

Construction of tame supercuspidal representations in arbitrary residue characteristic

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Pith's one-line read This paper constructs supercuspidal representations of tame reductive p-adic groups in every residue characteristic, including p=2, by compact induction from a Yu-type input that no longer needs Yu's second genericity condition; the…

desk verdict Genuine q>=4 construction of tame supercuspidals in residue characteristic 2; Theorem A's q>2 claim and the torsion-prime statements need correction before publication. read the letter →

arxiv 2501.18553 v2 pith:EDABEDVL submitted 2025-01-30 math.RT math.NT

Jessica Fintzen , David Schwein This is my paper · ORCID

classification math.RTmath.NT MSC 22E5011F27

keywords supercuspidalrepresentationsp-adicgroupsHeisenberg–WeilresiduecharacteristictwotameramificationcompactinductionMoy–PrasadfiltrationsCliffordtheory

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 paper aims to show that supercuspidal representations—the atomic building blocks of smooth representations of p-adic groups—can be constructed uniformly in every residue characteristic, including $p=2$. It extends Yu's 2001 construction, which required odd residue characteristic and two genericity conditions on its input, by removing the second genericity condition (GE2) and by repairing the characteristic-two mechanism that had blocked earlier work. The main theorem says that for any connected reductive group over a nonarchimedean local field that splits over a tamely ramified extension, any supercuspidal datum in the paper's sense yields an irreducible supercuspidal representation of $G(F)$ by compact induction, as long as the residue field has more than three elements. If the paper is right, the supercuspidal representations built by Yu—and in the classical setting all supercuspidal representations—are special cases of a single recipe that also covers the previously inaccessible $p=2$ case.

What carries the argument

The engine of the paper is the theory of Heisenberg $F_p$-groups and their Weil representations, rebuilt for $p=2$. A Heisenberg $F_p$-group is a finite $p$-group whose center has order $p$ and whose quotient by the center is an $F_p$-vector space; for odd $p$ these are the extraspecial $p$-groups of exponent $p$, while for $p=2$ there are two isomorphism classes of each order, built as central products of $D_8$ and $Q_8$. Each generic character $\varphi_i$ in the input produces a Heisenberg $F_p$-group $V_i^\natural$ inside a Moy–Prasad filtration quotient, and the Heisenberg representation $\omega_i$ of this group is the seed of $\kappa^-$. The new difficulty is the Weil representation: for $p=2$, the group $\mathrm{Aut}_Z(H)$ of center-fixing automorphisms of a Heisenberg group $H$ sits in a nonsplit extension $1\to F_2^{2n}\to \mathrm{Aut}_Z(H)\to O_{2n}(F_2)\to 1$, so the projective Weil representation cannot be linearized globally; the paper proves it linearizes over the particular subgroups arising from the p-adic group (Lemma 3.4.9) and pins down the linearization uniquely by requiring it to preserve the real or quaternionic structure of $\omega_i$, using the absence of order-two characters in the relevant quotient when $q>2$ (Lemma 4.5.5 and Proposition 4.5.6). The final supercuspidality argument rests on two small facts: if $|k|>3$, then for a quasi-split reductive $k$-group $H$ with parabolic $P$ and unipotent radical $U$ one has $U(k)\subseteq [P(k),U(k)]$ (Lemma 4.6.7), and a general group-theoretic observation that when $U\subseteq [P,U]$, a representation must act trivially on $U$ (Lemma 4.6.6).

What would settle it

For $k=F_3$, take $H=SL_2$ with its Borel subgroup $P$ and unipotent radical $U$: then $U(F_3)\cong F_3$ has order 3 while $[P(F_3),U(F_3)]$ is trivial, so Lemma 4.6.7 fails exactly at $q=3$ and the theorem excludes this case. To test the positive claim, carry out the construction of Example 4.4.2 for a 2-adic field with residue field $F_4$ and compute the intertwining algebra of the resulting compactly induced representation: Theorem 4.6.9(a) predicts it is one-dimensional, so any nonzero extra intertwiner would disprove the theorem.

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Extended reading notes

Core claim

The central theorem (Theorem 4.6.9(a)) asserts that, when the residue field $k_F$ has more than three elements, every supercuspidal $G$-datum $\Upsilon$—a chain of twisted Levi subgroups $G_1\supseteq\cdots\supseteq G_{n+1}$, a point $x$ in the Bruhat–Tits building, decreasing positive depths $r_i$, a depth-zero cuspidal representation $\rho$, and characters $\varphi_i$ that are generic of depth $r_i$ in the sense of (GE0) and (GE1) alone—yields an irreducible supercuspidal representation $\mathrm{c\text{-}ind}_{\widetilde K}^{G(F)}(\sigma)$. The subgroup $\widetilde K$ lies between Moy–Prasad-style subgroups $K$ and $K^+$, and $\sigma$ is built in two steps: a Heisenberg–Weil representation $\kappa^-$ of a smaller subgroup $K^-$ is constructed from Heisenberg $F_p$-groups $V_i^\natural$ extracted from the filtration quotients, and then Clifford theory extends $\kappa^-$ to $\kappa$ on $K$ and to $\sigma$ on $\widetilde K$. When $p$ is odd and every $\varphi_i$ satisfies Yu's second genericity condition (GE2), the intervening groups coincide, no choices are needed, and the construction specializes to Yu's; when $p=2$, the Heisenberg–Weil machinery is rebuilt on real and quaternionic representations, since the automorphism group of a Heisenberg $F_2$-group no longer splits as a symplectic semidirect product and its projective Weil representation has no global linearization. The paper also proves that $\widetilde K/K$ is a finite $p$-group, trivial under (GE2), and that the compact induction from $K$ decomposes as a direct sum of the new supercuspidal representations with positive multiplicities.

Load-bearing premise

The load-bearing premise is that the residue field $k_F$ has more than three elements: the final step of the supercuspidality proof needs the commutator fact that for a quasi-split reductive group over a field $k$ with $|k|>3$, the $k$-points of the unipotent radical of a parabolic subgroup lie in $[P(k),U(k)]$, and this fails for $k=F_3$.

Editorial extensions

If this is right

All supercuspidal representations constructed by Yu in 2001 are recovered as a special case; in particular, when $p$ does not divide the order of the absolute Weyl group of $G$, the new construction yields all supercuspidal representations of $G(F)$.
When every $\varphi_i$ satisfies (GE2), the subgroup $\widetilde K$ collapses to $K$ and $\mathrm{c\text{-}ind}_K^{G(F)}(\rho\otimes\kappa)$ is itself irreducible supercuspidal, with no Clifford choices required.
The construction works in residue characteristic two, where no general supercuspidal construction existed for arbitrary tame reductive groups, and it does so without invoking the Glauberman correspondence or Gérardin's Weil-representation analysis.
A single datum $\Upsilon$ can produce several non-isomorphic supercuspidal representations, corresponding to the Clifford-theoretic choices of $\kappa$ and $\sigma$; Example D.6 shows these choices can be forced and are detected by distinct formal degrees.

Reading between the lines

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

The Clifford-parameterized finite set attached to each datum looks like a natural bookkeeping device for the packet structure predicted by the local Langlands correspondence when the residue characteristic is small; the paper does not pursue this, but the framework is ready for it.
The real/quaternionic linearization technique should apply to other characteristic-two settings where classical Weil representations are unavailable, such as constructing types for classical groups or extending representations of unipotent radicals in small characteristic.
The natural next classification question is whether every tame supercuspidal representation arises from this construction; since it contains Yu's representations and drops (GE2), the answer may be yes in all tame settings, with the Clifford choices accounting for the finer structure.
If Lemma 4.6.7 could be sharpened to cover $|k|=3$, the construction itself would extend to residue fields of three elements, closing the gap between Theorem A's $q>2$ and the main theorem's $q>3$.

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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

Bottom line: this is a real advance, not a repackaging. Fintzen and Schwein build supercuspidal representations in residue characteristic two, remove Yu's second genericity condition, and do it with a uniform input. The Heisenberg–Weil theory over F2 with R-linearizations, the Clifford theory step for disconnected normalizers, and the avoidance of Gérardin's theorem are genuinely new and non-routine. The main theorem for q>3 is supported by a long, structured proof. I could not machine-check it, but the central argument looks sound to me, and the abstract's q>=4 restriction matches Theorem 4.6.9. The new pieces—the two isomorphism classes of Heisenberg F2-groups, the nonsplit sequence for Aut_Z(H) when n>=3, and the use of real/quaternionic structure to pin down the Weil representation—are real mathematics. Credit where due: the paper honestly states its boundary and does not pretend the q=3 case is handled inside the main proof.

Now the soft spots, in proportion. First, Theorem A states q>2, but the proof only gives q>3. The footnote delegates q=3 to a combination of [Yu01] and [Fin21], but does not show that those theorems accept the relaxed input without (GE2). If that delegation cannot be made rigorous, the q>2 claim in Theorem A is unsupported. The abstract is fine because it says q>=4; the discrepancy is between Theorem A and the rest of the paper. This needs to be fixed, not by hand-waving, but by proving the q=3 case or changing Theorem A.

Second, the torsion-prime implication is inverted in Section 4.1 and again in Theorem 4.6.9(c). GE2 is automatic when p is not a torsion prime for the dual group, not when it is. Example 4.1.3 itself illustrates this: an order-two character of an elliptic torus in SL2 at p=2, even though 2 is not a torsion prime for the SL2 root system (it is for PGL2). This is a genuine theorem-statement error, though it does not enter the q>3 proof. It will mislead readers and should be corrected.

The Lemma 4.6.7 boundary at |k|=3 is real and the authors know it; that is exactly why q=3 is excluded from Theorem 4.6.9. I do not see a hidden fatal flaw there—the proof genuinely stops at q>3, and the statements in the abstract match that.

Who is this for? Anyone working on p-adic groups, supercuspidal constructions, or the Yu machinery. It deserves a serious referee and a prominent venue once the statement-level issues are fixed. I would send it to review and expect a revise-and-resubmit.

Referee Report

2 major / 3 minor

Summary. The paper constructs tame supercuspidal representations of a connected reductive p-adic group G(F), attached to a 'supercuspidal G-datum' analogous to Yu's input but allowing residue characteristic 2 and omitting Yu's second genericity condition (GE2). The construction proceeds by compact induction from an open compact-mod-center subgroup, using a new Heisenberg-Weil theory for Heisenberg F_p-groups when p=2 and Clifford theory to handle the nonabelian normalizer that appears when (GE2) fails. The main theorem (Theorem 4.6.9) proves, for residue field cardinality q>3, that the resulting compactly induced representations are irreducible and supercuspidal. The abstract restricts to residue fields of size at least four, while the introduction's Theorem A claims q>2 and defers q=3 to a footnote.

Significance. If the q>3 theorem is correct, this is a substantial advance: it extends Yu's construction of tame supercuspidal representations to residual characteristic 2 and simultaneously removes Yu's second genericity condition. The char-2 Heisenberg-Weil linearization via real/quaternionic structures, the use of order-two character obstructions, and the Clifford-theoretic passage to a larger normalizer are genuinely new mechanisms. The paper is also commendably explicit about the q>3 boundary, and it includes useful appendices on commutator results and an explicit spin-group example. The cost is that two statement-level errors, discussed below, must be corrected before the paper can be accepted.

major comments (2)

[§1 (Theorem A), footnote 2; §4.6, Lemma 4.6.7 and Theorem 4.6.8(a)] The q>2 claim in Theorem A is not proved by the present arguments. Theorem 4.6.9 only proves the result for q>3, and the proof of Theorem 4.6.8(a) relies on Lemma 4.6.7, whose hypothesis |k|>3 is sharp: for H=SL_2 over F_3, every t in F_3^× satisfies t^2=1, so [S(k),U_α(k)] is trivial and U(k) is not contained in [P(k),U(k)]. The footnote's delegation of q=3 to a combination of [Yu01] and [Fin21] is not a proof that those references accept the relaxed input Υ without (GE2) at q=3; [Yu01] imposes (GE2), and no specific result of [Fin21] is cited that performs the relaxation. Since the abstract already states q≥4, the simplest fix is to state Theorem A for q>3 only, or to supply a genuine q=3 argument.
[§4.1 (p. 20), §4.3, and Theorem 4.6.9(c)] The torsion-prime implication is stated backwards in several places. The text says that generic characters satisfy (GE2) when p is a torsion prime for the Langlands dual group, and Theorem 4.6.9(c) repeats this; the correct statement is that (GE2) is automatic when p is not a torsion prime, as the paper itself states in the final sentence of Lemma 4.6.3 and as Example 4.1.3 demonstrates (p=2 is not a torsion prime for SL_2, yet the character constructed there fails (GE2)). This error does not enter the proof of Theorem 4.6.9(a), but it is a false assertion in the statement of part (c) and in the discussion in §4.3 of when the additional Clifford-theoretic choices are necessary.

minor comments (3)

[Theorem 4.6.9(a)] The displayed statement reads 'Let σ∈Irr(~K,K,ρ⊗σ)', but it should read 'ρ⊗κ'.
[§3.2, after Lemma 3.2.2] The sentence 'For our there is no need to assume that A or π have a particular form' is missing a word; it should read 'For our purposes'.
[References] Reference [Cot25] cites a MathOverflow post; consider replacing it with a published or more permanent source if one is available.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction in the central q>3 construction; Theorem A's q=3 delegation to [Yu01]+[Fin21] is an unverified non-circular gap.

full rationale

The central derivation is not circular. The input Υ is specified by (GE0)/(GE1) without (GE2), and the output c-ind_{~K}^{G(F)}(σ) is built from explicit Heisenberg–Weil theory (Section 3), the canonical representation κ^- of K^- (Lemma 4.5.8), and Clifford-theoretic choices for κ and σ (Section 4.3). The irreducibility and supercuspidality proof for q>3 (Theorem 4.6.8 and 4.6.9) is an intertwining argument: Theorem 4.6.8(a) shows that any intertwiner of σ lies in ~K, and Lemma B.1(c) converts this into irreducibility. The key new ingredients—Lemma 4.4.1 for Heisenberg F_p-groups at p=2, Lemma 3.4.9/Proposition 4.5.3 for linearizable Weil representations, and Lemmas 4.6.6–4.6.7—are proved in the paper or reduce to external results (Gérardin, Steinberg, Tits, Yu) that do not contain the target theorem. No fitted parameter is renamed as a prediction, and no equation defining κ^-, ~K, or σ is equivalent by construction to the claimed supercuspidality. The paper does rely on the authors' prior [Fin21] and [Fin] for the overall proof skeleton and for definitions, but the adaptations are written out, and the main q>3 claim does not reduce to a self-citation. Two non-circular concerns should be flagged explicitly. (i) Theorem A claims q>2, but Theorem 4.6.9 proves only q>3; the footnote on page 4 says 'Theorem 4.6.9 assumes q > 3, ultimately because of Lemma 4.6.7. When q = 3, although our analysis of the Heisenberg–Weil representation is insufficient to treat this case, one can combine [Yu01] and [Fin21] to construct supercuspidal representations from our slightly more general input Υ.' This delegation does not demonstrate that [Yu01] and [Fin21] accept the relaxed no-(GE2) input at q=3, so the q>2 statement is unsupported as written. This is a rigor/correctness gap, not a circular reduction. (ii) Section 4.1 and Theorem 4.6.9(c) invert the torsion-prime implication: GE2 is automatic when p is not a torsion prime, not when it is, and Example 4.1.3 itself exhibits a non-GE2 character at p=2 while 2 is not a torsion prime for the root system of SL_2. This statement error does not enter the q>3 proof. Because the central result is self-contained against external benchmarks, the circularity score stays in the 0–2 band; I assign 1 rather than 0 only to reflect the repeated, though non-load-bearing, reliance on the authors' own earlier framework.

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

No fitted constants or invented physical entities appear. The choices κ and σ in Section 4.3 are representation-theoretic choices, not numerical free parameters. The mathematical axioms used are standard background plus the explicit tameness and residue-field hypotheses of the theorem.

assumptions (5)

standard math Stone-von Neumann theorem for Heisenberg F_p-groups
Invoked in Lemma 3.3.1 to assert uniqueness and dimension of the Heisenberg representation.
standard math Galois descent, Hilbert 90, and Schur's lemma for linearizations
Used in Lemma 3.2.1 to show R-representation isomorphism classes are determined by complex representations.
domain assumption Bruhat-Tits theory, Moy-Prasad filtrations, and admissible embeddings of buildings
The whole construction is formulated in terms of Moy-Prasad subgroups; background is cited from [KP23].
domain assumption G splits over a tamely ramified extension and the residue field has q>3 for the main theorem
These are the stated hypotheses of the paper and Theorem 4.6.9.
standard math Steinberg's classification of torsion primes and Tits's commutator results for simply-connected quasi-split groups
Used in Lemma 4.1.2 and Appendix C to control disconnected centralizers and abelianizations over finite fields.

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Pith. "Pith review of Construction of tame supercuspidal representations in arbitrary residue characteristic." pith.science (2026). https://pith.science/paper/EDABEDVL

@misc{pith2026250118553,
  author       = {Pith},
  title        = {Pith review of: Construction of tame supercuspidal representations in arbitrary residue characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EDABEDVL}},
  note         = {Machine review of arXiv:2501.18553}
}

read the original abstract

Let F be a nonarchimedean local field whose residue field has at least four elements. Let G be a connected reductive group over F that splits over a tamely ramified field extension of F. We provide a construction of supercuspidal representations of G(F) via compact induction that contains, among others, all the supercuspidal representations constructed by Yu in 2001, but that also works in residual characteristic two. The input for our construction is described uniformly for all residual characteristics and is analogous to Yu's input except that we do not require our input to satisfy the second genericity condition (GE2) that Yu imposes.

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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

Green functions for positive-depth Deligne--Lusztig induction
math.RT 2025-06 accept novelty 7.0 of 10

For large q, positive-depth Deligne-Lusztig induction matches the Yu-Kaletha-FKS algebraic construction for all Howe-unramified elliptic pairs, via a characterization theorem and a Green function comparison.

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Adler, Reﬁned anisotropic K-types and supercuspidal representations , Paciﬁc Journal of Mathematics 185 (1998), no

[Adl98] Jeﬀrey D. Adler, Reﬁned anisotropic K-types and supercuspidal representations , Paciﬁc Journal of Mathematics 185 (1998), no. 1, 1–32. MR 1653184 [BK93] Colin J. Bushnell and Philip C. Kutzko, The admissible dual of GL(N ) via compact open sub- groups, Annals of Mathematics Studies, vol. 129, Princeton University Pre ss, Princeton, NJ,

work page 1998

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Winter, The automorphism group of an extraspecial p-group, Rocky Mountain J

MR 2562037 [Win72] David L. Winter, The automorphism group of an extraspecial p-group, Rocky Mountain J. Math. 2 (1972), no. 2, 159–168. MR 297859 [Yam22] Yuki Yamamoto, On Mackey decomposition for locally proﬁnite groups ,

work page 1972

[3] [166]

Howe, Tamely ramiﬁed supercuspidal representations of Gln, Paciﬁc J

MR 2431732 [How77] Roger E. Howe, Tamely ramiﬁed supercuspidal representations of Gln, Paciﬁc J. Math. 73 (1977), no. 2, 437–460. MR 0492087 [Kal19] Tasho Kaletha, Regular supercuspidal representations , J. Amer. Math. Soc. 32 (2019), no. 4, 1071–1170. MR 4013740 [Kal] , Supercuspidal L-packets. Preprint, available at https://arxiv.org/pdf/1912.03274v2.pd...

arXiv 1977

[4] [1980]

Griess Jr., Automorphisms of extra special groups and nonvanishing deg ree 2 cohomol- ogy, Paciﬁc J

MR 569209 [Gri73] Robert L. Griess Jr., Automorphisms of extra special groups and nonvanishing deg ree 2 cohomol- ogy, Paciﬁc J. Math. 48 (1973), 403–422. MR 476878 [HH56] P. Hall and Graham Higman, On the p-length of p-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42. MR 72872 [HM08] Jeﬀrey Hakim an...

work page 1973

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MR 794307 [Con14] Brian Conrad, Reductive group schemes, Autour des sch´ emas en groupes. Vol. I, 2014, pp. 93–444. MR3362641 [Cot25] Sean Cotner, What is the answer to the Kneser–Tits problem over a ﬁnite ﬁel d?,

work page 2014

[6] [1993]

MR1204652 [Bla93] Laure Blasco, Paires duales r´ eductives en caract´ eristique2, M´ em. Soc. Math. France (N.S.) 52 (1993), 1–73. MR 1212951 [Bou02] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6 , Elements of Mathematics, Springer- Verlag, Berlin,

work page 1993

[7] [2002]

MR 1890629 [BT65] Armand Borel and Jacques Tits, Groupes r´ eductifs, Inst

Translated from the 1968 French original by A ndrew Pressley. MR 1890629 [BT65] Armand Borel and Jacques Tits, Groupes r´ eductifs, Inst. Hautes ´Etudes Sci. Publ. Math. 27 (1965), 55–150. MR 207712 [Car85] Roger W. Carter, Finite groups of Lie type: Conjugacy classes and complex cha racters, Pure and Applied Mathematics, Wiley,

work page 1965

[8] [2005]

MR 2104929 [PR84] Gopal Prasad and M. S. Raghunathan, Topological central extensions of semisimple groups over local ﬁelds , Ann. of Math. (2) 119 (1984), no. 1, 143–201. MR 736564 [Ren10] David Renard, Repr´ esentations des groupes r´ eductifsp-adiques, Cours Sp´ ecialis´ es, vol. 17, Soci´ et´ e Math´ ematique de France,

work page 1984