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On depth-zero characters of p-adic groups

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Pith's one-line read Depth-zero characters of reductive p-adic groups admit one uniform description, even when the group is wildly ramified.

desk verdict Solid torus results and a needed proof of Langlands' bijection, but the main theorem's proof has a characteristic-p gap in Proposition 3.3. read the letter →

arxiv 2502.01505 v1 pith:MNU5PUZ4 submitted 2025-02-03 math.NT math.RT

Maarten Solleveld , Yujie Xu This is my paper · ORCID
classification math.NTmath.RT MSC 11S3722E5020G25
keywords localLanglandscorrespondencedepth-zerocharactersp-adicgroupswildlyramifiedMoy–PrasadfiltrationsBruhat–Titsbuildingtoriparahoricsubgroups
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 establishes that, for any connected reductive group over a non-archimedean local field, the several natural definitions of a depth-zero character all agree, with no tameness assumption on the group. A character trivial on the simply connected cover of the derived group has depth zero on every maximal torus exactly when it kills the pro-unipotent radical of every parahoric subgroup, and checking one suitably chosen torus or one Iwahori subgroup is enough. Along the way, the authors prove that the local Langlands correspondence for arbitrary tori restricts to an isomorphism between depth-zero characters and depth-zero Langlands parameters, and that Langlands' natural map from $H^1(W_F, Z(G^\vee))$ to characters of $G/G_{sc}$ is bijective. These results make depth-zero characters a uniformly accessible class, which the authors use as input for local Langlands correspondences for depth-zero representations.

What carries the argument

The central mechanism is the surjectivity of the product map $T'_r \times G_{sc,f,r} \to G_{f,r}$ at level $r = 0+$ (Proposition 3.3), proved for every facet $f$ of the Bruhat–Tits building. It is obtained by passing to the associated graded Moy–Prasad isomorphisms $G_{f,r}/G_{f,s} \cong \mathfrak{g}_{f,r}/\mathfrak{g}_{f,s}$ and checking surjectivity of the Lie-algebra addition map $\mathfrak{t}'(\mathbb{F}^{nr})_r \oplus \mathfrak{g}_{der}(\mathbb{F}^{nr})_{f,r} \to \mathfrak{g}(\mathbb{F}^{nr})_{f,r}$ on the unramified splitting, then transferring the surjectivity back to the group level via $p$-adic completeness. For the torus results, the load-bearing identity is Proposition 2.1: the LLC for tori sends the norm map $N_{E/F}$ to restriction $\operatorname{Res}^{W_F}_{W_E}$, which makes the depth-zero restriction follow from Pontryagin duality and the structure of $I_F/P_F$.

What would settle it

The claim would be refuted by exhibiting a wildly ramified torus $T$ and a smooth character $\chi$ trivial on $T_{0+}$ whose Langlands parameter is nontrivial on the wild inertia group $P_F$, or by a reductive group $G$ with a character $\chi$ trivial on $G_{sc}$ that satisfies one of the four depth-zero conditions and violates another.

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Extended reading notes

Core claim

For an arbitrary torus $T$ over a non-archimedean local field $F$, the local Langlands correspondence restricts to an isomorphism $$\operatorname{Hom}(T/T_{0+},\mathbb{C}^\times) \cong $H^{1}$(W_F/P_F, T^\vee{}^{P_F}),$$ identifying characters trivial on the pro-$p$ radical $T_{0+}$ of the parahoric subgroup $T_0$ with cocycles trivial on wild inertia $P_F$. For any connected reductive group $G$, a character $\chi$ trivial on the image of the simply connected cover $G_{sc}$ has depth zero on every maximal torus if and only if its kernel contains the pro-unipotent radical $G_{f,0+}$ of every parahoric subgroup; one Iwahori subgroup or one maximal torus containing a maximal unramified torus already suffices. The proof also establishes that Langlands' map $H^1(W_F, Z(G^\vee)) \to \operatorname{Hom}(G/G_{sc},\mathbb{C}^\times)$ is a natural topological isomorphism.

Load-bearing premise

The argument assumes that a certain technical comparison between the filtration of the group and that of its Lie algebra, originally proved for tamer settings, remains valid when the group is wildly ramified; if that comparison fails anywhere, the equivalence between the torus and parahoric tests for depth zero breaks down.

Editorial extensions

If this is right

  • For every connected reductive group over a non-archimedean local field, tensoring a depth-zero representation by a character in $X_0(G)$ preserves depth zero, so the depth-zero category is stable under such twists.
  • The depth-zero part of the LLC for tori is now available for arbitrary tori: $\operatorname{Hom}(T/T_{0+},\mathbb{C}^\times) \cong H^1(W_F/P_F, T^\vee{}^{P_F})$ even when $T$ splits only over a wildly ramified extension.
  • Langlands' map $H^1(W_F, Z(G^\vee)) \to \operatorname{Hom}(G/G_{sc},\mathbb{C}^\times)$ is a natural isomorphism of topological groups for all connected reductive groups over local fields.
  • The equivalence of the four depth-zero conditions reduces checking depth zero to a single maximal torus containing a maximal unramified torus, which is the practical criterion for applications.
  • For separable extensions, the character associated to the restriction of a parameter is the norm-twisted character, a naturality property not previously recorded.

Reading between the lines

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

  • Beyond the paper, the same equivalence of depth-zero conditions might extend to characters not assumed trivial on $G_{sc}$, provided the anisotropic part of the derived group is handled separately; the authors do not claim this.
  • Beyond the paper, Theorem 3.1 implies that the full character group of $G/G_{sc}$ is governed by $H^1(W_F, Z(G^\vee))$, and Lemma 3.2 identifies the depth-zero part in the tame case; whether the equality $X_0(G^\vee) = H^1(W_F/P_F, Z(G^\vee)^{P_F})$ holds for wildly ramified groups is not settled here.
  • Beyond the paper, Proposition 2.1 invites a base-change statement for depth-zero LLC: restriction of a parameter to $W_E$ should correspond to the norm of the character, which could give a purely local route to cyclic base change for depth-zero representations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper studies the local Langlands correspondence (LLC) for arbitrary tori over local fields and gives a detailed analysis of depth-zero characters of reductive p-adic groups, allowing wildly ramified groups. The main results are: (i) a norm-compatibility property for the LLC for tori (Proposition 1.2/2.1); (ii) a depth-zero restriction of the LLC for arbitrary tori (Proposition 1.3/2.2); (iii) a proof of bijectivity of Langlands' map H^1(W_F, Z(G^∨)) → Hom(G/G_sc, C^×) (Theorem 3.1); and (iv) several equivalent characterizations of depth-zero characters of G, defined either by restriction to maximal tori or by triviality on pro-unipotent radicals of parahoric subgroups (Theorems 3.4 and Lemma 3.5, summarized as Theorem 1.4). The paper is written for non-archimedean local fields of arbitrary characteristic, with the wildly ramified case highlighted as the main difficulty.

Significance. If the results are correct, the paper provides a clean and uniform description of depth-zero characters of arbitrary reductive p-adic groups, removing tameness hypotheses that appear in earlier work. Proposition 2.2 is a genuinely new statement for wildly ramified tori, and Theorem 3.1 fills a gap in the literature by proving bijectivity of Langlands' homomorphism. The equivalence of the torus and parahoric notions of depth zero is an important structural result that is likely to be useful in the authors' companion work on the LLC for depth-zero representations. The proofs are explicit, use commutative diagrams, and carefully distinguish new content from cited results. The main weakness is a gap in Proposition 3.3 for positive characteristic, which currently leaves Theorem 1.4 unsupported in that setting.

major comments (1)
  1. [§3, Proposition 3.3, diagram (3.10)–(3.12)] The proof states 'The Lie algebra of T′ × Gsc is t′ ⊕ gder' immediately before (3.9). This identification is valid only when the differential of the central isogeny q: Gsc → Gder is surjective, which fails for inseparable isogenies in positive characteristic. For example, for G = PGL_p over F_p((t)), the image of dq: Lie(SL_p) → Lie(PGL_p) is the derived subalgebra of Lie(PGL_p), a proper subspace of codimension 1. Consequently, the lower row of diagram (3.10) does not represent the tangent space of the domain (T′_r × Gsc,f,r)/(T′_s × Gsc,f,s), and the surjectivity of the addition map (3.11) for t′ ⊕ gder does not, as written, imply the surjectivity of the left vertical map (3.12). Since Proposition 3.3 is the load-bearing step for Theorem 3.4 and Lemma 3.5, and since Theorem 1.4 is claimed for every non-archimedean local field without a characteristic-zero assumption, the proof is incomplete in positive characteristic. The authors should either add a characteristic-zero hypothesis to Theorem 1.4 or supply a separate argument for inseparable isogenies verifying that t′(Fnr)_r + dq(Lie(Gsc))(Fnr)_{f,r} = g(Fnr)_{f,r} for the appropriate graded pieces.
minor comments (4)
  1. [§3, Theorem 3.1, equation (3.1)] The proof invokes the existence of a z-extension (3.1) with H^1(F,D) = 1 without giving a proof or a precise reference. This is a standard construction (e.g., via induced tori R_{E/F} G_m and Hilbert 90), but a citation would help the reader, especially since the argument relies on it.
  2. [Abstract] There is a typographical error in the abstract: 'a rbi-trary' should be 'arbitrary'.
  3. [§2, Proposition 2.1, proof of (2.2)–(2.3)] In the proof of Proposition 2.1, representatives ̅γ for cosets in W_F/W_K are used in formulas (2.2) and (2.3) without an explicit statement that such representatives are chosen once and for all; a short clarifying sentence would improve readability.
  4. [§3, Theorem 3.4(b)] The phrase 'for one chamber C' could be misread as 'for a single, unspecified chamber'; the intended meaning is 'for some chamber C' or 'for any chamber C', since the proof shows all chambers are equivalent by transitivity of G on chambers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations rely on external structural results and newly defined equivalences, not on self-citation or fitted inputs.

full rationale

Walked the derivation chain. Proposition 2.2 derives depth-zero preservation for arbitrary tori from the existing LLC for tori (Theorem 1.1), the weakly unramified isomorphism (1.2), Pontryagin duality, and a pro-p-quotient argument on H^1(I_F/P_F, T∨, P_F); the target isomorphism is not assumed as an input. Theorem 3.1 proves bijectivity of Langlands' map using the LLC for tori, the exact sequence (3.5), and Kottwitz's connectedness result; no fitted or renamed parameter appears. The load-bearing Proposition 3.3 rests on Kaletha–Prasad's Moy–Prasad isomorphism [KaPr, Theorem 13.5.1] and root-space generation of the associated graded Lie algebra; these are external published results, not self-citations, and the proposition is used to prove, rather than to define, the equivalence in Theorem 3.4. Self-citations [SoXu1, SoXu2] appear only as motivation and applications, not as premises. The skeptical concern about the Lie algebra identification t′⊕gder in positive characteristic is a possible mathematical correctness gap, not a circular reduction: no equation of the paper is assumed in order to derive itself. Under the hard rules, a gap without an exhibited reduction to the paper's own inputs does not constitute circularity. Therefore the circularity score is 0.

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

No free parameters and no speculative entities are introduced. The groups X0(G) and X0(G∨) are notation for sets, not new mathematical objects with independent evidence burdens. All premises are citable theorems or standard constructions in arithmetic geometry; the most delicate input is the Kaletha-Prasad Bruhat-Tits machinery.

assumptions (7)
  • standard math Local Langlands correspondence for tori: Hom(T(F), C×) ≅ H^1(WF, T∨), functorial in tori and extending local class field theory.
    Invoked throughout (Propositions 2.1, 2.2, Theorem 3.1, Lemma 3.5) as established background from Langlands, Borel, and Yu.
  • standard math Pontryagin duality identifies Hom(T0+, C×) as the maximal pro-p quotient of Hom(T0, C×), and IF/PF has no nontrivial pro-p quotients (NSW 7.5.2).
    Used in Proposition 2.2 to convert the depth-zero condition into absence of nontrivial pro-p quotients.
  • standard math Kottwitz and Tate-Nakayama isomorphisms: π0((D∨)^{WF}) ≅ Hom(H^1(F,D), C×), and H^1(F, Gsc) = 1 for local fields.
    Used in Theorem 3.1 to prove connectedness of (D∨)^{WF} and to obtain the bijection from the long exact sequence.
  • standard math Existence of a z-extension 1 -> D -> Gtilde -> G -> 1 with D an F-torus with H^1(F,D) = 1 and Gsc embedding in Gtilde.
    Chosen in the proof of Theorem 3.1; the existence is a standard construction in the field but not proved in the paper.
  • standard math Kaletha-Prasad Bruhat-Tits theory: Moy-Prasad filtrations, the associated graded isomorphism [KaPr, Theorem 13.5.1], and transitivity of Gsc on apartments.
    Core to Proposition 3.3, which is used to derive the parahoric characterization of X0(G) in Theorem 3.4.
  • standard math Lang-Steinberg theorem: G is quasi-split over the maximal unramified extension, and the centralizer of a maximal unramified torus Tnr is a maximal torus.
    Used in Lemma 3.5 to reduce the universal torus condition to a single well-chosen torus.
  • standard math Functoriality of torus filtrations under norm maps, as stated in [KaPr, Definition 7.2.2 and Proposition B.10.10].
    Used in Lemma 3.5 to transfer the depth-zero property from F to an unramified splitting extension E.

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Cite this review

Pith. "Pith review of On depth-zero characters of p-adic groups." pith.science (2026). https://pith.science/paper/MNU5PUZ4

@misc{pith2026250201505,
  author       = {Pith},
  title        = {Pith review of: On depth-zero characters of p-adic groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MNU5PUZ4}},
  note         = {Machine review of arXiv:2502.01505}
}
read the original abstract

We show new properties of the Langlands correspondence for arbitrary tori over local fields. Furthermore, we give a detailed analysis of depth-zero characters of reductive p-adic groups, for groups that may be wildly ramified. We present several different definitions of ``depth-zero'' for characters, and show that these notions are in fact equivalent. These results are useful for proving new cases of local Langlands correspondences, in particular for depth zero representations.

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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

  1. Formal degree of principal series of quasi-split groups

    math.RT 2026-04 unverdicted novelty 6.0 of 10

    The formal degree conjecture holds for discrete series representations inside principal series of quasi-split groups over local fields, via type construction and the local Langlands correspondence from prior work.

Reference graph

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