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On parameters of Hecke algebras for $p$-adic groups

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Pith's one-line read Every depth-zero type's affine Hecke algebra is isomorphic to a unipotent type's.

desk verdict A careful reduction of q-parameters to unipotent ones; worth refereeing, with the main caveat being its reliance on two external Jordan-decomposition compatibility results. read the letter →

arxiv 2505.16040 v3 pith:DPT2CM5J submitted 2025-05-21 math.RT math.NT

Kazuma Ohara This is my paper
classification math.RTmath.NT MSC 22E5020C0820C33
keywords p-adicgroupsHeckealgebrastypesunipotentrepresentationsBernsteinblocksaffineq-parametersdepth-zero
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 establishes that the affine Hecke algebra attached to any depth-zero type of a connected reductive p-adic group is isomorphic to the affine Hecke algebra attached to a unipotent type for an associated connected reductive group that splits over an unramified extension. Because Hecke algebras attached to types govern Bernstein blocks, this reduces the calculation of the previously unknown parameters of these algebras—dimension ratios of induced representations—to the explicitly known unipotent case. The proof compares a numerical invariant, the q-parameter of a length-two representation, across finite-field parabolic inductions and then lifts the equality to p-adic intertwiners. A conjecture on parameters for arbitrary Bernstein blocks follows under the assumption that the group splits over a tamely ramified extension and the residue characteristic does not divide the order of the absolute Weyl group.

What carries the argument

The load-bearing object is the q-parameter: for a finite-dimensional representation of length two, the ratio $\dim(\pi_1)/\dim(\pi_2)$ of its constituents. Affine Hecke algebra parameters $q_s$ are exactly such ratios, via the standard identification of q-parameters with dimension ratios of parabolically induced finite reductive representations. The argument shows these ratios are invariant under two moves compatible with the construction of types: passing through a group homomorphism with abelian cokernel (Proposition 2.1), and passing from a cuspidal representation to the unipotent representation attached to it by the Jordan decomposition over a finite field (Proposition 3.2.3). To transfer the equality to p-adic groups, the paper constructs a connected reductive group $G_\theta$ from the $\theta$-orthogonal affine root system of the original datum, normalizing the affine roots by positive scalars so that they form a genuine affine root system, and proves the parahoric quotients of $G$ and $G_\theta$ are adjointly isomorphic up to duals; this makes the unipotent comparison valid even when $G$ is ramified.

What would settle it

Compute directly the dimension ratio of the two constituents of the induced representation for a non-unipotent cuspidal representation of a maximal Levi subgroup of $\mathrm{GL}_3$ over the field of two elements; if it differs from the unipotent-side ratio, the finite-field equality on which Theorem 4.4.1 relies fails.

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Extended reading notes

Core claim

The central claim is Theorem 4.4.1: for a depth-zero type $(K, \rho)$ of a connected reductive p-adic group $G$, the sets of $K$-relevant affine hyperplanes and the affine Weyl groups they generate coincide with the corresponding objects for a unipotent type $(K_\theta, u)$ of an associated group $G_\theta$ that splits over an unramified extension. The parameters agree on simple reflections, so the affine Hecke algebras $H_C(W(\rho_M)_{\mathrm{aff}}, q)$ and $H_C(W(u_{\rho_M})_{\mathrm{aff}}, q_\theta)$ are isomorphic. Since earlier work had already identified the full Hecke algebra attached to any tame type with a depth-zero Hecke algebra, the same isomorphism holds for the affine part of Hecke algebras attached to tame types. Under the assumptions that $G$ splits over a tamely ramified extension and the residue characteristic does not divide the order of the absolute Weyl group, this proves the conjecture that the parameters of the Hecke algebra attached to an arbitrary Bernstein block coincide with those of a unipotent Bernstein block.

Load-bearing premise

The calculation assumes that the dimension ratio of an induced representation survives unchanged when the representation is replaced by its unipotent counterpart through the Jordan decomposition in the relevant finite reductive groups.

Editorial extensions

If this is right

  • The q-parameters of any depth-zero affine Hecke algebra can now be read off from the explicit unipotent parameter lists rather than computed hyperplane by hyperplane.
  • The same equality holds for the affine Hecke algebra factors attached to tame types, so the only unknown part of those Hecke algebras is the twisted group algebra factor.
  • For groups splitting over a tamely ramified extension with residue characteristic not dividing the absolute Weyl group order, every Bernstein block has a Hecke algebra whose parameters equal those of a unipotent block.
  • The isomorphism preserves the standard anti-involutions, so the involutive structure of the affine Hecke algebra is also captured by the unipotent model.
  • The earlier principal-series identification of such Hecke algebras with Iwahori–Hecke algebras appears as the special case where the depth-zero datum is induced from a torus.

Reading between the lines

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

  • The paper proves the isomorphism only for the affine Hecke algebra factor; if the twisted group algebra factors were also shown to be isomorphic, the entire depth-zero Hecke algebra would be Morita equivalent to a unipotent one.
  • The construction of $G_\theta$ by normalizing affine roots suggests that, for ramified groups, $G_\theta$ is the correct dual-side object for a Bernstein block; identifying it in explicit examples would test whether the comparison extends to the full block.
  • Because the q-parameter comparison is essentially a finite-field computation, the same route could yield other invariants of depth-zero Bernstein blocks—for instance formal degrees—once the full Hecke algebra isomorphism is in hand.
  • The scaling factors used in the normalization enter only through the construction of $G_\theta$; computing them in one ramified example would show whether the unipotent comparison is genuinely needed there.
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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

0 major / 7 minor

Summary. This paper proves that for a depth-zero type (K, ρ) of a connected reductive p-adic group G, the affine Hecke algebra H(W(ρ_M)_{aff}, q) appearing in the structure theorem of [AFMO24a, Thm 5.3.6] is isomorphic to the affine Hecke algebra attached to a unipotent type for a group G_θ that splits over an unramified extension (Theorem 4.4.1). The key finite-field ingredient is Proposition 3.2.3, which equates the q-parameter of the parabolic induction of a cuspidal representation with that of a unipotent cuspidal representation via Jordan decomposition. Combining this with [AFMO24b] yields Theorem 4.5.1 and, under the usual tameness assumptions, a version of Lusztig's conjecture on parameters of Hecke algebras (Theorem 4.6.2). The proof is detailed, with explicit reduction steps in Sections 2 and 3.3.

Significance. If correct, the main result gives an explicit reduction of the parameters of arbitrary Bernstein Hecke algebras to the unipotent case, where they are already described by Lusztig. This is a substantial step beyond previous results of Roche and others. The proof is coherent; the main fragility is the reliance on [GM20, Cor 4.7.6 and 2.6.6] for the finite-field q-parameter equality. However, the paper's reduction in Section 3.3 explicitly reduces to the connected-center, absolutely-simple case, which is the hypothesis these results require, so this is an external dependency rather than an internal inconsistency. The paper also honestly acknowledges the non-canonical choices in the disconnected-center case (Remark 3.2.2).

minor comments (7)
  1. [Title page] The title page reads "forp-adic groups"; it should read "for p-adic groups".
  2. [Section 2, proof of Proposition 2.1] The text "irreducible constitutes" should be "irreducible constituents".
  3. [Section 3.4] Please provide the precise statements (with theorem numbers or page references) of [GM20, Corollary 4.7.6] and [GM20, Corollary 2.6.6] as used in the proof of Proposition 3.2.3, including the hypotheses on the group; the reduction in Section 3.3 is designed to meet them, but the reader cannot verify this without the exact statements.
  4. [Section 4.3, Theorem 4.3.7] The notation is inconsistent: the representation is introduced as u_x, but the theorem states q_{θ,s} = q(ind_{K_{θ,h}}^{K_{θ,x}}(ρ_{θ,x})); please unify the notation.
  5. [Section 3.2, paragraph before Proposition 3.2.3] The grammar "the equivalence relations ∼ is defined" should be "the equivalence relation ∼ is defined".
  6. [Section 3.3, Remark 3.3.1] The phrase "a prioridepends" should be "a priori depends".
  7. [Section 4.4, proof of Theorem 4.4.1] The notation "H K-rel" appears without subscripts; it should be H_{K-rel} for consistency with the surrounding text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the q-parameter equality is proved from external finite-group Jordan decomposition theorems, not from the target isomorphism or from fitted parameters.

full rationale

The central claim (Theorem 4.4.1) reduces to the finite-field q-parameter equality Proposition 3.2.3, q(R^G_M(ρ_M)) = q(R^{G*_s}_{M*_s}(u*_M)). The q-parameter is defined independently as the dimension ratio of the two irreducible constituents of a length-two parabolically induced representation; it is not defined in terms of the unipotent side, and it is not fitted to the conclusion. The equality is obtained from [GM20, Cor. 4.7.6] (Jordan decompositions commute with parabolic induction) and [GM20, Cor. 2.6.6] (preservation of dimension ratios), both external to the present paper. The reductions in Section 3.3 use Proposition 2.1 and regular embeddings, again without presupposing the desired equality. The construction of G_θ and u_{M,x0} in Section 4.3 chooses the unipotent representation via the Jordan decomposition and adjoint isomorphisms, but the equality of the q-parameters of the induced representations, and hence the Hecke algebra parameters q_s = q_{θ,s}, is a theorem (Corollary 3.5.6) rather than a definitional identification. The self-citations [AFMO24a, AFMO24b] provide the structural description of the depth-zero Hecke algebra and the reduction from Kim-Yu types to depth zero; neither of those statements is the target equality, so their use is load-bearing but not circular. Remark 3.3.1 openly records the dependency on [GM20] and the need for connected center; this is a caution about external assumptions, not a circular step. No step in the derivation is equivalent to its input by construction, and no prediction is a renamed fitted parameter.

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

No free parameters are fitted. The paper introduces the group G_theta as an explicit construction, not as a postulated entity. All assumptions are standard results in the field or explicit hypotheses of the theorems.

assumptions (7)
  • standard math Bernstein decomposition and the theory of types (existence of s-types) [Ber84, BK98]
    Used in Section 1 to frame the Hecke algebra attached to a Bernstein block.
  • domain assumption Kim-Yu construction yields types for every Bernstein block when G splits over a tamely ramified extension and p does not divide |W| [KY17, Fin21b]
    Assumed in Sections 4.5 and 4.6 to cover arbitrary Bernstein blocks.
  • domain assumption Explicit description of depth-zero Hecke algebras as semidirect products [AFMO24a, Theorem 5.3.6]
    The paper's starting point in Section 4.2; from the authors' previous preprint.
  • domain assumption Lusztig's Jordan decomposition exists and commutes with parabolic induction under connected-center/simple conditions, preserving dimension ratios [Lus88, GM20 Cor 4.7.6, Cor 2.6.6]
    Core input for Prop 3.2.3 and Cor 3.5.6.
  • standard math Bruhat-Tits theory: parahoric subgroups, reductive quotients, affine root systems [BT72, KP23]
    Used throughout Section 4 to define parahoric subgroups and G_theta.
  • standard math Vanishing of H^1 and H^2 of Gal(F^unr/F) on T_ad(F^unr)_0 [DR09, Kal19a]
    Used in Section 4.3 to lift the 1-cocycle defining the F-structure of G_theta.
  • standard math Finiteness and semisimplicity of length at most two induced representations over finite fields
    Used in the definition of q-parameters and Mackey formula arguments in Section 3.

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Pith. "Pith review of On parameters of Hecke algebras for $p$-adic groups." pith.science (2026). https://pith.science/paper/DPT2CM5J

@misc{pith2026250516040,
  author       = {Pith},
  title        = {Pith review of: On parameters of Hecke algebras for $p$-adic groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DPT2CM5J}},
  note         = {Machine review of arXiv:2505.16040}
}
abstract

Let $F$ be a non-archimedean local field with residue characteristic $p$ and $G$ be a connected reductive group defined over $F$. In earlier joint works with Jeffrey D. Adler, Jessica Fintzen, and Manish Mishra, we proved that the Hecke algebras attached to types constructed by Kim and Yu are isomorphic to the Hecke algebras attached to depth-zero types. Note that if $G$ splits over a tamely ramified extension of $F$ and $p$ does not divide the order of the absolute Weyl group of $G$, such Hecke algebras cover the Hecke algebras attached to arbitrary Bernstein blocks. We also proved that for a depth-zero type $(K, \rho)$, the corresponding Hecke algebra $\mathcal{H}(G(F), (K, \rho))$ has an explicit description as a semi-direct product of an affine Hecke algebra $\mathcal{H}(W(\rho_M)_{\mathrm{aff}}, q)$ with a twisted group algebra $\mathbb{C}[\Omega(\rho_{M}), \mu]$, generalizing prior work of Morris. In this paper, we show that the affine Hecke algebra $\mathcal{H}(W(\rho_M)_{\mathrm{aff}}, q)$ appearing in the description of the Hecke algebra $\mathcal{H}(G(F), (K, \rho))$ attached to a depth-zero type $(K, \rho)$ is isomorphic to the one attached to a unipotent type for a connected reductive group splitting over an unramified extension of $F$. This makes it possible to calculate the parameters of the affine Hecke algebras for depth-zero types and types constructed by Kim and Yu explicitly. In particular, we prove a version of Lusztig's conjecture that the parameters of the Hecke algebra attached to an arbitrary Bernstein block agree with those of a unipotent Bernstein block under the assumption that $G$ splits over a tamely ramified extension of $F$ and $p$ does not divide the order of the absolute Weyl group of $G$.

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