Stability of Elliptic Fargues-Scholze $L$-packets

Chenji Fu

REVIEW 2 major objections 3 minor 34 references

Stability of Elliptic Fargues-Scholze $L$-packets

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that every irreducible representation attached to an elliptic Fargues–Scholze L-parameter yields a nonzero stable combination of Harish-Chandra characters, establishing stability of elliptic L-packets by geometric methods.

desk verdict Strong, original proof of stability for elliptic FS L-packets, endoscopy-free and uniform in G, but the standing assumption of an F-rational Borel containing T_g fails for anisotropic tori and is load-bearing. read the letter →

arxiv 2501.00652 v1 pith:LAJQFCGY submitted 2024-12-31 math.RT math.AGmath.NT

Chenji Fu This is my paper · ORCID

classification math.RTmath.AGmath.NT MSC 22E5011R3914D24

keywords Fargues-ScholzeL-packetsellipticL-parametersstableHarish-ChandracharacterslocalLanglandscorrespondencegeometrizationspectralactionweightmultiplicitiesp-adicreductivegroups

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

Stability of $L$-packets is a central prediction of the local Langlands correspondence: the Harish-Chandra characters of members of a packet should admit a linear combination invariant under stable conjugacy. This paper proves that prediction for every elliptic $L$-parameter, using the Fargues–Scholze geometrization of representations as sheaves on the stack of $G$-bundles over the Fargues–Fontaine curve. For each irreducible $\pi$ with Fargues–Scholze $L$-parameter $\varphi$, the paper constructs a finite virtual representation $\pi_0$ by acting on $\pi$ with the regular representation of the centralizer $S_\varphi$ modulo the fixed center, and shows that the alternating sum of its Harish-Chandra characters is stable on elliptic regular semisimple elements. In characteristic zero the same object is a nonzero stable distribution on $G(F)$. The interest is a new route to stability that does not use endoscopic classification and works in positive characteristic.

What carries the argument

The load-bearing construction is the spectral action of Fargues–Scholze: perfect complexes on the stack of $L$-parameters act on the derived category of sheaves on $\mathrm{Bun}_G$, and the averaged object $(i_\varphi)_* O(S_\varphi/Z(\widehat G)^\Gamma) * (i_1)_! \pi$ is shown to be a Hecke eigensheaf, so $T_{V_\mu}$ multiplies it by $\dim V_\mu$. The Hansen–Kaletha–Weinstein formula rewrites the resulting character identity as a weighted sum over stable conjugacy classes, with weights $\dim V_\mu[\lambda]/\dim V_\mu$ indexed by an invariant in the finite abelian group $H_g = \ker(X_*(T_g)_\Gamma \to \pi_1(G)_\Gamma)$. Choosing $\mu = 4m\rho_G$ and applying the Weyl character formula, the paper proves that these weight multiplicities equidistribute over $H_g$ as $m \to \infty$; Fourier analysis on $H_g$ then makes the weighted sum independent of the class of $g'$, which is exactly stability.

What would settle it

For $G = \mathrm{GL}_2(\mathbb{Q}_p)$, take an elliptic regular element $g$ whose centralizer is a quadratic-field torus. Compute the difference $S_{h,m} - S_{h',m}$ for the two classes $h,h'$ in $H_g$: the proof requires this to tend to $0$, so a nonzero limit would falsify the equidistribution step. Separately, in a case where the Fargues–Scholze packet is known, compare $\Theta_{\pi_0}$ on two stably conjugate elliptic elements; unequal values would refute the stability conclusion.

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Extended reading notes

Core claim

Let $G$ be a connected reductive group over a non-archimedean local field $F$, and let $\varphi: W_F \to \widehat G(\overline{\mathbb Q}_\ell)$ be an elliptic $L$-parameter. For every irreducible smooth representation $\pi$ of $G(F)$ whose Fargues–Scholze $L$-parameter is $\varphi$, the paper defines $F_0 = (i_\varphi)_* O(S_\varphi/Z(\widehat G)^\Gamma) * (i_1)_! \pi$ and $\pi_0 = i_1^* F_0$. It proves that $\pi_0$ is a finite direct sum of irreducible representations (up to degree shifts) containing $\pi$, and that the Harish-Chandra character $\Theta_{\pi_0}$ is a nonzero function on the elliptic regular semisimple locus $G(F)_{\mathrm{ell}}$ invariant under $G(\overline F)$-conjugacy. In characteristic zero $\Theta_{\pi_0}$ is a nonzero stable distribution on all of $G(F)$. This establishes the stability of the Fargues–Scholze $L$-packet $\Pi^{\mathrm{FS}}_\varphi(G)$ in the sense required by the stability conjecture, without invoking the theory of endoscopy.

Load-bearing premise

The proof assumes that for every elliptic regular element $g$ the centralizer torus $T_g$ has an $F$-rational Borel subgroup containing it; for non-split elliptic tori this can fail, and without a chosen Borel the weight multiplicities cannot be canonically compared across the stable conjugacy class.

Editorial extensions

If this is right

Every elliptic Fargues–Scholze packet, whenever nonempty, carries a nonzero stable character combination, so the stability part of the local Langlands stability conjecture holds for these packets.
The stable combination is canonically built from any member: $\Theta_{\pi_0}$ for $\pi_0 = O(S_\varphi/Z(\widehat G)^\Gamma) * \pi$, with coefficients coming from an equal-weight limit of weight multiplicities.
The method is independent of endoscopic classification and covers positive characteristic, where full endoscopy is not available.
The same weighted-sum identity transfers character values between extended pure inner forms, up to the sign $(-1)^{\langle \mu, 2\rho_G\rangle}$.
The equidistribution of weight multiplicities (Theorem 4.3.2) is a separate, self-contained result about highest-weight representations of reductive groups.

Reading between the lines

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

If the Fargues–Scholze packet is nonempty for every elliptic parameter, the same construction would prove the stability conjecture for all elliptic discrete-series packets; the paper leaves nonemptiness open.
The regular representation $O(S_\varphi/Z(\widehat G)^\Gamma)$ may be the correct canonical packet average; in cases where a classical packet is known, this stable combination should agree with the classical stable packet character.
A testable extension is to compute $S_{h,m}$ explicitly for a small-rank split group and a non-split elliptic torus, to measure how quickly the equidistribution limit is approached.

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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

Dear Colleague,

The stress-test note holds up. This paper has a genuine gap in a load-bearing place; everything around it is as strong as the reader says. Fu proves that every irreducible π whose Fargues–Scholze L-parameter is elliptic lies in a finite set of representations with the same FS parameter whose Harish-Chandra characters admit a non-zero integer combination stable under G(F)-conjugacy on elliptic regular semisimple elements. That is the first stability result for FS L-packets, it is endoscopy-free, uniform in G, and works in positive characteristic. If the gap is fixed, this is an important paper.

Credit where due. The Hecke eigensheaf observation (Prop 4.1.2) — that (i_φ)_* O(S_φ/Z(Ĝ)^Γ) ∗ (i_1)!π is a Hecke eigensheaf — is clean and will be reused. The reduction of stability to equidistribution of weight multiplicities via HKW22 is a genuinely new route, and the Fourier analysis on the finite group H_g (Prop 3.3.1, Thm 4.3.1) is elegant and self-contained. The paper is honest about its limits: Remark 2.4.12 says the constructed set may be a proper sub-packet, Remark 4.3.4 says existence of π with a given parameter is not known in general, Section 1.2 says compatibility with classical LLC is open. No circularity; the spectral action and HKW22 are external and do not include the target theorem. The char-0 distribution statement is derived from Arthur, standard.

Now the soft spot, and it is real. Sections 1.1, 4.2 and 4.3 choose an F-rational Borel containing the elliptic maximal torus T_g. For the unramified quadratic torus in GL_2(Q_p) — the centralizer of an elliptic element — no such Borel exists, and for non-quasi-split G none exists at all. The identification X_*(T_g) ≅ X_*(T_univ), the comparison of inv(g,g') with weights λ, and the weighted-sum formula Cor 4.2.8 all depend on that choice. The proof as written does not reach basic anisotropic cases. It looks repairable — work over F-bar with a fixed Borel and track the non-equivariant identification, or use the canonical identification of X_*((T_g)_sc) with the coroot lattice — but the repair is not written down, and the Γ-twist may change the equidistribution input.

One framing note: the title says stability of FS L-packets, while the theorem produces a stable sub-packet containing each π. The abstract is precise about this; just do not over-read the title.

This is for people working in geometric local Langlands and p-adic representation theory; the equidistribution section is worth reading on its own. The paper deserves a serious referee: the ideas are original, the theorem is significant, and the gap, while central, is plausibly fixable. Send it out, and ask the referee to check the Borel assumption on the GL_2 elliptic case first.

Best, [You]

Referee Report

2 major / 3 minor

Summary. The paper proves stability of a virtual Harish-Chandra character attached to an elliptic Fargues-Scholze L-parameter. For an elliptic L-parameter ϕ and an irreducible smooth representation π of G(F) with Fargues-Scholze parameter ϕ, the author defines π0 := O(S_ϕ / Z(Ĝ)^Γ) * π via the spectral action, shows that the corresponding sheaf on Bun_G is a Hecke eigensheaf, and uses the Hansen-Kaletha-Weinstein character formula to express Θ_{π0}(g) as a weighted sum over the stable conjugacy class of an elliptic element g. The proof then reduces stability to an equidistribution statement for weight multiplicities of V_{μ_m}, proved by Fourier analysis on the finite abelian group H_g. The main theorem asserts that Θ_{π0} is stable on G(F)_ell and, in characteristic zero, is a non-zero stable distribution.

Significance. If correct, this provides a new, endoscopy-independent proof of stability for Fargues-Scholze L-packets, with the advertised advantage of working in positive characteristic. The argument is genuinely constructive and has no fitted parameters: the Hecke eigensheaf property is proved in Proposition 4.1.2, the equidistribution statement is proved in Theorem 4.3.1, and the use of the spectral action and HKW22 as external benchmarks avoids circularity. The main defect is a repeated false geometric assumption about the existence of an F-rational Borel subgroup containing an elliptic maximal torus; this affects the formulation of the weighted-sum formula and the equidistribution theorem, and therefore the proof of the main theorem as written.

major comments (2)

[§1.1 Step 2; §4.2 before Corollary 4.2.8; §4.3 before Theorem 4.3.1] The proof repeatedly assumes that for every elliptic g ∈ G(F)_ell there exists a Borel subgroup defined over F containing T_g = Cent(g,G). This assumption is false in general. For example, in G=GL_2 over a non-archimedean local field, an element whose centralizer is the unramified quadratic torus is elliptic, but an F-rational Borel subgroup contains only split maximal tori. Consequently, the identification X_*(T_g) ≅ X_*(T_univ) with a distinguished dominance order, the definition of H_g, and the reindexing λ=inv(g,g') in Corollary 4.2.8 are not justified for such elements. Since the weighted-sum formula (19) and the quantities S_{h,m} in Theorem 4.3.1 are the inputs to the stability proof, Theorem 4.3.3 is not proven for elliptic elements with anisotropic centralizer as written. The text supplies no alternative construction for groups or elements where the assumption fails.
[§3.2, Lemma 3.2.2] The surjection Λ^Φ → H_g used in Proposition 3.3.1(2) is constructed via a non-canonical isomorphism X_*((T_g)_sc) ≅ Λ^Φ that is explicitly noted not to be Γ-equivariant. The paper does not prove that the resulting character χ of Λ^Φ, and hence the existence of β with χ(β) ≠ 1, is independent of this choice. While the conclusion is likely true because any two choices differ by a Weyl-group element and the weight multiplicities of V_{μ_m} are W-invariant, this independence is not stated or proved; as written, the growth estimate depends on choices that are not shown to be canonical.

minor comments (3)

[Throughout] Several typographical errors should be fixed: "a prior" should be "a priori" (e.g., in §1.2 and Remark 4.3.4), "Combing Corollary 4.2.8" should be "Combining", and "charater" in Lemma 4.3.5 should be "character".
[Introduction, after Equation (9)] The line "π0 = i_1^*F0 F0 ≅ i_1!π0" appears garbled; it should read "π0 := i_1^*F0, and F0 ≅ i_1!π0".
[§1.1, Step 2] The phrase "We choose a Borel subgroup over F containing (T_g)_F" is not just notationally strong but mathematically impossible for anisotropic elliptic tori; as noted in the major comments, this needs to be replaced by an admissible embedding or by a Borel over an algebraic closure with an explicit independence statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability theorem is derived from the external spectral action, the HKW22 transfer formula, and the Weyl character formula, none of which assume the target result.

full rationale

The paper's derivation chain is self-contained against external benchmarks rather than circular. The object π0 is defined by the spectral action of the regular representation O(Sϕ) on π, and the Hecke eigensheaf property (Proposition 4.1.2) is proved from the projection formula and the semisimplicity of the regular representation, not from stability. The weighted-sum formula (Corollary 4.2.8) is obtained by combining this eigensheaf property with the Hansen–Kaletha–Weinstein transfer formula [HKW22, Theorem 6.5.2], which is an independent external result about local shtuka spaces. The equidistribution step (Theorem 4.3.1 and Theorem 4.3.2) is a genuine asymptotic statement about weight multiplicities of highest weight representations, proved via the Weyl character formula and Fourier analysis on the finite abelian group Hg; no fitted parameter or hidden input is renamed as a prediction. The theorem's conclusion — stability of Θπ0 — is not assumed at any point; it is derived after the limit of the coefficients is shown to be independent of the representative g′. The paper explicitly notes in Section 1.2 that it proves something different from classical endoscopy-based results, and no load-bearing self-citation chain appears: the main cited inputs [FS21] and [HKW22] are external and do not include the stability theorem. The skeptical concern about the assumption of an F-rational Borel containing T_g is a mathematical correctness issue about a genuinely needed hypothesis, not a circularity issue, because the assumption does not encode the conclusion. Therefore no circular step is present, and the score is 0.

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

The central claim relies on a large external machinery: Fargues-Scholze's geometrization (spectral action, Hecke operators, L-parameter stack), Hansen-Kaletha-Weinstein's trace formula for local shtuka spaces, Harish-Chandra character theory, and Arthur's stabilization. These are treated as black boxes. No free parameters are introduced and no new entities are postulated.

assumptions (5)

domain assumption Fargues-Scholze spectral action exists and is compatible with Hecke operators (FS21, Corollary X.1.3).
Used to define pi0 and prove the Hecke eigensheaf property (Prop 4.1.2, Lemma 4.2.2).
domain assumption Hansen-Kaletha-Weinstein trace formula (Theorem 2.6.1) computes the Harish-Chandra character of the Hecke correspondence.
Establishes Equation (18) and the weighted-sum expression (Cor 4.2.8).
domain assumption Harish-Chandra character theory (local integrability, linear independence) holds over Q_l and in positive characteristic.
Needed to define Theta_pi0 and to deduce non-vanishing from linear independence (Lemma 4.3.5).
domain assumption Arthur's stabilization theorem (Art96, Theorem 6.1) for elliptic virtual characters.
Upgrades stability on elliptic elements to a stable distribution in characteristic zero (Theorem 4.3.6).
standard math Standard structure of reductive groups over local fields: Borovoi fundamental group, Kottwitz set, elliptic tori.
Background for H_g, B(T_g), and the Fourier argument.

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Cite this review

Pith. "Pith review of Stability of Elliptic Fargues-Scholze $L$-packets." pith.science (2026). https://pith.science/paper/LAJQFCGY

@misc{pith2026250100652,
  author       = {Pith},
  title        = {Pith review of: Stability of Elliptic Fargues-Scholze $L$-packets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LAJQFCGY}},
  note         = {Machine review of arXiv:2501.00652}
}

abstract

Let $F$ be a non-archimedean local field. Let $\overline{F}$ be an algebraic closure of $F$. Let $G$ be a connected reductive group over $F$. Let $\varphi$ be an elliptic $L$-parameter. For every irreducible representation $\pi$ of $G(F)$ with Fargues--Scholze $L$-parameter $\varphi$, we prove that there exists a finite set of irreducible representations $\{\pi_i\}_{i \in I}$ containing $\pi$, such that $\pi_i$ has Fargues--Scholze $L$-parameter $\varphi$ for all $i \in I$ and a certain non-zero $\mathbb{Z}$-linear combination $\Theta_{\pi_0}$ of the Harish-Chandra characters of $\{\pi_i\}_{i \in I}$ is stable under $G(\overline{F})$ conjugation, as a function on the elliptic regular semisimple elements of $G(F)$. Moreover, if $F$ has characteristic zero, $\Theta_{\pi_0}$ is a non-zero stable distribution on $G(F)$.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

On local character relations

James Arthur. On local character relations. Sel. Math., New Ser. , 2(4):501--579, 1996

work page 1996

[2] [2]

The endoscopic classification of representations

James Arthur. The endoscopic classification of representations. Orthogonal and symplectic groups , volume 61 of Colloq. Publ., Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2013

2013

[3] [3]

Adler and Loren Spice

Jeffrey D. Adler and Loren Spice. Supercuspidal characters of reductive \(p\) -adic groups. Am. J. Math. , 131(4):1137--1210, 2009

work page 2009

[4] [4]

Gonz \'a lez-Avil \'e s

Mikhail Borovoi and Cristian D. Gonz \'a lez-Avil \'e s. The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme. Cent. Eur. J. Math. , 12(4):545--558, 2014

work page 2014

[5] [5]

Galois action on B orovoi's algebraic fundamental group

Mikhail Borovoi (https://mathoverflow.net/users/4149/mikhail borovoi). Galois action on B orovoi's algebraic fundamental group. MathOverflow. url: https://mathoverflow.net/q/484122 (version: 2024-12-15)

work page 2024

[6] [6]

Abelian Galois cohomology of reductive groups , volume 626 of Mem

Mikhail Borovoi. Abelian Galois cohomology of reductive groups , volume 626 of Mem. Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 1998

work page 1998

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Affine Springer fibers and depth zero L-packets

Roman Bezrukavnikov and Yakov Varshavsky. Affine Springer fibers and depth zero L -packets. Preprint, arXiv :2104.13123 [math. RT ] (2021), 2021

work page Pith review arXiv 2021

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Bill Casselman. Introduction to admissible representations of p-adic groups. unpublished notes , 1995. url: https://www.math.utah.edu/ ptrapa/math-library/casselman/casselman-p-adic-book.pdf

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