arXiv subjects

Arnaud Eteve

Publications and source records attributed to Arnaud Eteve.

8 recordsLinked to original sources

arXiv2026

The extended Fargues--Scholze spectral action

Let $G$ be a connected reductive group over a non-archimedean local field. The main result of Fargues and Scholze [FS21] for the geometrization of the local Langlands correspondence is the construction of a ``spectral action'' on the category of $\ell$-adic sheaves on $\text{Bun}_{G}$, the stack of $G$-torsors on the Fargues-Fontaine curve. The goal of this paper is to prove a conjecture of Fargues which says that one can extend this construction to the larger stack $\text{Bun}_G^e$ of $G$-torsors on the Kaletha gerbe over the curve, as introduced by Fargues [Far22]. This ``extended spectral action'' allows for a version of the categorical local Langlands conjecture for an arbitrary connected reductive group $G$, and is the first such statement for those $G$ which are not extended pure inner forms of a quasi-split group, such as non-trivial inner forms of $\mathrm{SL}_{n}$. Finally, we prove this conjecture for tori, following the original argument of [Zou24] and, under the same assumptions as [Zou26] (including connected center), we reduce the ``extended'' version of the categorical conjecture to the one in Fargues--Scholze.

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arXiv2025

The Parabolic K-motivic Hecke Category

We define and study the parabolic K-motivic Hecke category of a (possibly disconnected) Kac-Moody group. Our main result is a combinatorial description via singular K-theory Soergel bimodules which arise from the equivariant algebraic K-theory of parabolic Bott-Samelson resolutions. In the spherical affine case, the K-motivic Hecke category serves as one side of a conjectural quantum K-theoretic derived Satake equivalence, addressing a conjecture of Cautis-Kamnitzer.

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arXiv2025

Universal Koszul Duality for Kac-Moody Groups

We prove a monoidal equivalence, called universal Koszul duality, between genuine equivariant K-motives on a Kac-Moody flag variety and constructible monodromic sheaves on its Langlands dual. The equivalence is obtained by a Soergel-theoretic description of both sides which extends results for finite-dimensional flag varieties by Taylor and the first author. Universal Koszul duality bundles together a whole family of equivalences for each point of a maximal torus. At the identity, it recovers an ungraded version of Beilinson-Ginzburg-Soergel's and Bezrukavnikov-Yun's Koszul duality for equivariant and unipotently monodromic sheaves. It also generalizes Soergel-theoretic descriptions for monodromic categories on finite-dimensional flag varieties by Lusztig-Yun, Gouttard and the second author. For affine Kac-Moody groups, our work sheds new light on the conjectured quantum Satake equivalences by Cautis-Kamnitzer and Gaitsgory. On our way, we establish foundations on six functors for reduced K-motives and introduce a formalism of constructible monodromic sheaves.

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arXiv2025

Tilting representations of finite groups of Lie type

Let $\mathbf{G}$ be a connected reductive group over a finite field $\mathbb{F}_q$ of characteristic $p > 0$. In this paper, we study a category which we call Deligne--Lusztig category $\mathcal{O}$ and whose definition is similar to category $\mathcal{O}$. We use this to construct a collection of representations of $\mathbf{G}(\mathbb{F}_q)$ which we call the tilting representations. They form a generating collection of integral projective representations of $\mathbf{G}(\mathbb{F}_q)$. Finally we compute the character of these representations and relate their expression to previous calculations of Lusztig and we then use this to establish a conjecture of Dudas--Malle.

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arXiv2025

Free monodromic Hecke categories and their categorical traces

The goal of this paper is to give a new construction of the free monodromic categories defined by Yun. We then use this formalism to give simpler constructions of the free monodromic Hecke categories and then compute the trace of Frobenius and of the identity on them. As a first application of the formalism, we produce new proofs of key theorems in Deligne--Lusztig theory.

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arXiv2025

Jordan decomposition, categorical traces and Deligne-Lusztig theory

This paper is a continuation of the author's previous work. We use the categorical trace formalism to construct a categorical Jordan decomposition for representations of finite groups of Lie type. As a second application, we study the endomorphism algebra of the Gelfand-Graev representation and recover a result of Li and Shotton-Li.

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arXiv2024

Nearby cycles commute with proper direct image on stacks of shtukas

Let $G$ be a generically reductive group over a smooth projective curve $X$ over a finite field. For any finite set $I$, we show that nearby cycles commute with proper direct image from stacks of shtukas to $X^I$. This generalizes some results of Salmon and the authors.

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arXiv2022

Kazhdan-Laumon sheaves and Deligne-Lusztig representations

Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.

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