Branching rules for irreducible depth-zero supercuspidal representations of $\mathrm{SL}(2,F)$, when $F$ has residual characteristic $2$

Monica Nevins; Zander Karaganis

Branching rules for irreducible depth-zero supercuspidal representations of $\mathrm{SL}(2,F)$, when $F$ has residual characteristic $2$

T0 review · 0 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper gives the complete decomposition of the restriction to a maximal compact subgroup of every depth-zero supercuspidal representation of SL(2,F) when F has residual characteristic 2.

desk verdict This paper closes the last open rank-one branching case for depth-zero supercuspidals on SL(2,F), and the central argument holds up despite a few cosmetic slips. read the letter →

arxiv 2509.01843 v1 pith:XGNROLNB submitted 2025-09-01 math.RT

Zander Karaganis , Monica Nevins This is my paper · ORCID

classification math.RT MSC 22E50

keywords branchingrulessupercuspidalrepresentationsresidualcharacteristic2SL(2F)nilpotentorbitslocalcharacterexpansionMackeytheoryp-adicgroups

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 completes the branching rules for depth-zero supercuspidal representations of SL(2,F) when F is a 2-adic field or a Laurent series field of characteristic two. It proves that restricting such a representation to a maximal compact subgroup yields a direct sum of irreducible components whose number and depth are governed by square classes modulo successive congruence subgroups. Each positive-depth component is explicitly constructed as an induced representation from a degenerate nilpotent coset, so the branching rules are controlled by the geometry of nilpotent orbits. The same machinery yields a representation-theoretic analogue of the local character expansion that remains meaningful in characteristic two, where the classical character expansion is unavailable.

What carries the argument

The Mackey components σ(ℓ)=Ind_{B'_ℓ}^{K'} g_ℓ σ, obtained from Hansen's GL(2) branching rules, are the objects to decompose. Their self-intertwining dimension is computed in Theorem 5.5 as |S_{⌈ℓ/2⌉}| using Corollary 3.6, a dyadic arithmetic classification of pairs a,d ≡ 1 mod P with a≡d mod P^δ and ad≡1 mod P^ℓ. The irreducible pieces I(ζ,u,ℓ) are built by Clifford theory from the character η_{(u,ℓ)} of the congruence subgroup K'_{⌈ℓ/2+⌉}, extended to the subgroup Γ(ℓ)', whose size sharply reflects the dichotomy at ℓ=4e.

What would settle it

Enumerate, for F=Q2 and δ=2, ℓ=4, all pairs (a,d)∈(1+P)² with a≡d mod P² and ad≡1 mod P⁴, and check that (a−d) mod P³ is always the same single residue; the theory predicts exactly one class. Finding two distinct residues would refute Corollary 3.6 and change the number of components at depth ℓ.

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Extended reading notes

Core claim

The central result (Theorem 7.12) gives the complete branching rules for any irreducible depth-zero supercuspidal representation π of SL(2,F) with F of residual characteristic 2. If π has no fixed vectors under K'_+, its restriction to the maximal compact subgroup K' is a direct sum, over odd depths ℓ, of |S_{(ℓ+1)/2}| irreducible components I(1,u,ℓ); otherwise π is induced from a cuspidal σ of the finite group SL(2,f) and the restriction is σ plus a direct sum, over even positive depths, of |S_{ℓ/2}| components. Each component I(1,u,ℓ) is constructed in (7.8) as an induction from a character attached to the degenerate (−ℓ,−ℓ/2) coset of the nilpotent element Xuϖ^{−ℓ}. The index set S_m cons

Load-bearing premise

The load-bearing premise is a specific arithmetic fact about squaring in the field: for units congruent to 1, the residue of (a−d) in the quotient P^δ/P^{δ+1} follows a four-case pattern, with the delicate case δ=2e giving a half-size image because the map x↦ιx+x² has a two-element kernel.

Editorial extensions

If this is right

For F of residual characteristic 2, the restriction of any depth-zero supercuspidal representation to K' is now explicitly known, component by component, for every depth.
The number of irreducible components at depth ℓ is |S_{⌈ℓ/2⌉}|, which stabilizes at 2qe for ℓ ≥ 4e+1 in the 2-adic case and grows without bound when char(F)=2.
The representations I(ζ,u,ℓ) provide an explicit family of irreducible K'-representations of prescribed depth arising from nilpotent orbits, extending the p-odd picture.
The local character expansion has a representation-theoretic version (Theorem 8.6) holding on K'_{4e+1} for 2-adic fields, and a family of local expansions (Theorem 8.10) valid in all characteristics, including char F=2.
The dimension of the largest irreducible component in π^{K'_n} grows at a rate that can be computed exactly, exhibiting a half-factor at the boundary ℓ=4e.

Reading between the lines

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

The same σ(ℓ)-breaking method may apply to positive-depth supercuspidals of SL(2,F), with the finite cuspidal σ replaced by a Bushnell–Kutzko type; the I(ζ,u,ℓ) family is expected to dominate those branching rules too.
The dyadic arithmetic lemma (Corollary 3.6) is a statement about the field's squaring map, so the same four-case dichotomy should control intertwining in other rank-one groups in residual characteristic 2.
When q=2, the Hecke-algebra computation in Section 6 suggests that End_{K'}(σ(ℓ)) is isomorphic to the group algebra of a cyclic group of order |S_{⌈ℓ/2⌉}| for ℓ ≥ 4e+1; verifying this in general would give an explicit basis of intertwiners.
The 'close cousins' phenomenon—nilpotent orbits collapsing into one degenerate coset—might offer a way to formulate orbital integrals in characteristic 2, connecting to germ-expansion analogues.

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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

Karaganis and Nevins have done it: the complete branching rules for depth-zero supercuspidal representations of SL(2,F) when F has residual characteristic 2. This case was explicitly excluded in all earlier work, including Nevins's own, and the paper delivers not just a decomposition but an explicit construction of every irreducible component from degenerate nilpotent cosets. That is a real result, not a translation of the p-odd arguments.

The strongest part is the Section 7 construction. The representations I(1,u,ℓ) are built concretely, their irreducibility and distinctness are proved via Mackey theory, and Theorem 7.12 states the branching formula cleanly. The growth of the component count with ℓ in characteristic 2 is a genuinely new phenomenon, and the local-character-expansion analogues in Section 8 are a thoughtful payoff. The dyadic arithmetic in Section 3 is intricate; my spot checks of Lemma 3.1, Proposition 3.5, and Corollary 3.6 came out consistent, including the q/2 image in the δ=2e case.

The soft spots are all cosmetic. Theorem 1.3 says "2qe + 1" summands while the body consistently gives 2qe; Remark 8.7 claims e=1 in the p-odd comparison when the correct value is e=0; and the Section 6 commutator argument is a bit soft, but Section 6 is explicitly not used later. None of these touches the main theorem.

I also want to note the stress-test concern about the dyadic classification. It is true that Corollary 3.6 is a delicate load-bearing piece for the intertwining results of Section 5. But the branching formula of Theorem 7.12 does not actually depend on that classification: the index set S_{⌈ℓ/2⌉} comes from the square-class lemma, and Corollary 7.11 gets the decomposition by direct Mackey construction. So even if a subcase of Proposition 3.5 needed adjustment, the central formula would survive. That is reassuring.

This paper deserves a serious referee. It is important, detailed, and the core argument is credible. The referee's job is to check the Section 7 Mackey calculations and clean up the small typos, not to reopen the main conclusion. I would accept it.

Referee Report

0 major / 4 minor

Summary. The paper determines the complete branching rules for restriction to a maximal compact subgroup K' = SL(2,R) of every irreducible depth-zero supercuspidal representation of G' = SL(2,F) when F has residual characteristic 2. The main result (Theorem 7.12) gives an explicit multiplicity-one decomposition: either π = Ind_{K'}^{G'} σ and Res_{K'} π = σ ⊕ ⊕_{ℓ even} ⊕_{u∈S_{ℓ/2}} I(1,u,ℓ), or π has no K'_+-fixed vectors and Res_{K'} π = ⊕_{ℓ odd} ⊕_{u∈S_{(ℓ+1)/2}} I(1,u,ℓ). The irreducible components I(1,u,ℓ) are constructed by induction from characters attached to degenerate (−ℓ,−ℓ/2) cosets of nilpotent elements. The proof combines Mackey theory, a detailed dyadic arithmetic analysis of intertwining operators (Sections 3 and 5), and a new construction of irreducible K'-representations from nilpotent orbits (Section 7). Applications include representation-theoretic analogues of the local character expansion (Section 8).

Significance. This is a substantial advance: it completes the depth-zero supercuspidal branching problem for SL(2) in the previously inaccessible residual-characteristic-2 case. The paper reveals a genuinely new phenomenon—the number of irreducible components at fixed depth grows without bound when char(F)=2—in contrast to the constant multiplicity for odd p. It also gives explicit, geometric constructions of all components in terms of nilpotent orbits and formulates local character expansion analogues. The main proof is detailed and internally consistent; on spot checks the cardinalities, dimension bookkeeping, and the independence of the branching formula from the dyadic intertwining classification all hold. The explicit constructions and finiteness statements are important assets.

minor comments (4)

[§8.2.2] The displayed isomorphism Res_{K_1} π ≅ (1−q)1 ⊕ Res_{K_1} τ_GL(O,1) appears to have the wrong sign. By Theorem 4.1, Res_{K_1} π = (q−1)1 ⊕ Res_{K_1} τ_GL(O,ω), and Res_{K_1} J(ω,ℓ) is independent of ω because ω is trivial on K_1. Thus the scalar should be (q−1)1, not (1−q)1. If a Grothendieck-group identity with a different formal coefficient is intended, please state this explicitly.
[Theorem 7.9] The double-coset representatives for Γ(ℓ)\K/Γ(ℓ) and Γ(ℓ)'\K'/Γ(ℓ)' are stated without proof ('following a similar strategy to Proposition 5.3'). Since these parametrizations control irreducibility and the square-class indexing, please add a short proof or a precise lemma.
[Theorem 1.2 / Theorem 8.6] The equality in Theorem 8.6 is only valid in the Grothendieck group, since nπ is negative; please include this qualification in the theorem statement and in Theorem 1.2 in the introduction.
[Theorem 8.10] The notation π^{K'_ℓ} is used as a direct summand but has not been defined as a subrepresentation; earlier in §8.1 it denotes the fixed subspace. Since K'_ℓ is normal, the fixed subspace is a K′-subrepresentation; please state this explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main branching formula is proven by independent Mackey/Clifford constructions, not by its own inputs.

full rationale

The derivation chain for the central result, Theorem 7.12, is self-contained. The decomposition of each Mackey component σ(ℓ) is established in Corollary 7.11 using Proposition 7.10 (which identifies σ(ℓ) with Res_{K'} J(ω,ℓ)) and Theorem 7.9, which independently constructs irreducible representations I(1,u,ℓ) via Clifford theory and a double-coset calculation for Γ(ℓ)'\K'/Γ(ℓ)'. The index set S_{⌈ℓ/2⌉} is fixed by square-class arithmetic in Lemma 3.1, not by the intertwining-dimension count; Corollary 5.6 is a consistent check (dim End_{K'}(σ(ℓ)) = |S_{⌈ℓ/2⌉}|), not an ingredient in Corollary 7.11. The dyadic classification in Proposition 3.5 and Corollary 3.6 is an independent statement about squaring in local fields of residual characteristic 2 and does not assume any branching conclusion. No parameter is fitted to a subset of data and renamed a prediction: the representations I(1,u,ℓ) are constructed before the decomposition, and the equality in Corollary 7.11 is proven by Frobenius reciprocity and Mackey theory. Self-citations to the authors' earlier p-odd papers are used for comparison and as the p-odd special case, not as load-bearing justification for the p=2 result. The coefficient nπ in Theorem 8.6 is solved from the proven isomorphism (equation 8.1), not used to force it; negative coefficients are normal in local character expansions and do not indicate circularity. The paper even explicitly notes, in Section 8.2.1, that it does not know a reference for wavefront sets when p=2 and proposes a convention; this is an acknowledged gap in external support, not a circular derivation.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The paper's results rest on standard machinery of p-adic representation theory (Moy-Prasad depth/filtrations, Mackey theory, Deligne-Lusztig theory) plus its own dyadic arithmetic. No numeric parameters are fitted: q, e, and the index sets S_m are structural data of the field and explicitly computed; the only set cardinalities that matter (|S_k| = q^{⌊k/2⌋} for k ≤ 2e, |S| = 2qe) are derived in Lemma 5.2 and Section 3. The new mathematical objects are the representations I(ζ,u,ℓ), J(ζ,ℓ), τ(O), τ_{ζ,u,ℓ}; they are constructed explicitly (eqs. (7.7), (7.8), Definitions 8.3, 8.9) rather than postulated, and their key properties (irreducibility, isomorphism classes, degrees) are proven in Theorems 7.9 and Lemma 8.1.

assumptions (7)

domain assumption Moy-Prasad theory: existence of depth, filtrations G_{x,r}, the Moy-Prasad isomorphism (2.1), and the classification of depth-zero supercuspidals by compact induction from cuspidal representations ([MP96, Prop 6.6]).
Invoked in Sections 2.3 and 4.1-4.2 to set up the representations π0(σ), π1(σ) and the notion of depth used throughout.
domain assumption Character table of GL(2,F_q) and the fact that for q even every cuspidal representation restricts irreducibly to SL(2,F_q) ([DM91, Ch.15]).
Used in Section 4 and Lemma 5.4 to compute trace characters χ_ℓ and the intertwining values q−1, −1, 0.
standard math Hansen's branching theorem for GL(2,F) (Theorem 4.1, [Han87]).
Black box giving Res_K π for depth-zero supercuspidals of GL(2,F); the paper reuses its Mackey coset structure and marks it as valid for all residual characteristics.
standard math Mackey's theorem in Kutzko's compact-induction form (Proposition 2.1, [Kut77]).
Foundation of the decomposition into σ(ℓ) in Section 4 and of all intertwining computations in Sections 5 and 7.
standard math The structure of R^×/(R^×)^2 for dyadic fields (Lemma 3.1, from [Cas23] in char 0 and direct computation in char 2).
Gives the representatives S and the count 2qe (char 0) or infinity (char 2) that feed the index sets S_m.
standard math Clifford theory for finite quotients (Theorem 2.2).
Used in Corollary 4.4 (equal degrees of components) and Theorem 7.9 (irreducibility via normalizers).
domain assumption Existence of Harish-Chandra local character expansion when char(F)=0 ([HC99]).
Context for Section 8.2; Theorem 8.6 is framed as a representation-theoretic analogue of this expansion, and the paper notes no such expansion is known in char 2 (nilpotent orbital integrals may not converge).

invented entities (3)

I(ζ,u,ℓ): irreducible representations of K' = SL(2,R) of depth ℓ, induced from characters attached to degenerate (−ℓ,−ℓ/2) cosets of nilpotent elements. independent evidence
purpose: The irreducible pieces in the branching rules (Theorem 7.12) and building blocks of the local character expansion.
Explicitly constructed in (7.8); irreducibility, distinctness up to square classes mod P^m, and degrees are proven in Theorem 7.9 and Lemma 8.1, all checkable independently of the branching-rule theorem.
J(ζ,ℓ): irreducible representations of GL(2,R) attached to nilpotent cosets. independent evidence
purpose: Recovers Hansen's depth-ℓ Mackey components of GL(2,F) supercuspidals (Proposition 7.10); used for the GL(2) local expansion (Section 8.2.2).
Constructed in (7.7); proven isomorphic to Ind^K_{B_ℓ} g_ℓ σ via an intertwining computation.
τ(O,ζ) and τ_{ζ,u,ℓ}: (infinite) direct sums of I(ζ,u,ℓ) over depths, attached to nilpotent orbits and to degenerate cosets respectively. independent evidence
purpose: Gives the representation-theoretic local character expansion (Theorems 8.6, 8.10) including the characteristic-2 case.
Definitions 8.3 and 8.9 specify the summands explicitly; the paper's Q2 example (8.4) gives the concrete identity Res = −41·1 + Σ τ(O), providing a checkable instance.

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Pith. "Pith review of Branching rules for irreducible depth-zero supercuspidal representations of $\mathrm{SL}(2,F)$, when $F$ has residual characteristic $2$." pith.science (2026). https://pith.science/paper/XGNROLNB

@misc{pith2026250901843,
  author       = {Pith},
  title        = {Pith review of: Branching rules for irreducible depth-zero supercuspidal representations of $\mathrmSL(2,F)$, when $F$ has residual characteristic $2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XGNROLNB}},
  note         = {Machine review of arXiv:2509.01843}
}

abstract

We give the decomposition into irreducible representations of the restriction to a maximal compact subgroup of any irreducible depth-zero supercuspidal representation of $\mathrm{SL}(2,F)$ when $F$ is a local nonarchimedean field of residual characteristic two. We furthermore provide explicit constructions of these irreducible components in terms of nilpotent orbits, proving a representation-theoretic analogue of the local character expansion that holds even in the wild case of characteristic two.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Dan Barbasch and Allen Moy, Local character expansions, Ann. Sci. \'Ecole Norm. Sup. (4) 30 (1997), no. 5, 553--567. 1474804

work page 1997

[2] [2]

William Casselman, The restriction of a representation of G L _2(k) to G L _2( O ) , Math. Ann. 206 (1973), 311--318

work page 1973

[3] [3]

, Local quadratic extensions, Preprint https://personal.math.ubc.ca/ cass/research/pdf/QuadraticCFT.pdf, 2023

work page 2023

[4] [4]

Raf Cluckers, Julia Gordon, and Immanuel Halupczok, Local integrability results in harmonic analysis on reductive groups in large positive characteristic, Ann. Sci. \' E c. Norm. Sup\' e r. (4) 47 (2014), no. 6, 1163--1195. 3297157

work page 2014

[5] [5]

Stephen Debacker, Homogeneity results for invariant distributions of a reductive p -adic group , Ann. Sci. \' E cole Norm. Sup. (4) 35 (2002), no. 3, 391--422. 1914003

work page 2002

[6] [6]

Stephen DeBacker, Parametrizing nilpotent orbits via B ruhat- T its theory , Ann. of Math. (2) 156 (2002), no. 1, 295--332. 1935848

work page 2002

[7] [7]

21, Cambridge University Press, Cambridge, 1991

Fran c ois Digne and Jean Michel, Representations of finite groups of L ie type , London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991

work page 1991

[8] [8]

Kristina Hansen, Restriction to GL _2( O ) of supercuspidal representations of GL _2(F) , Pacific J. Math. 130 (1987), no. 2, 327--349. 914105

work page 1987