arXiv subjects

Jean-François Dat

Publications and source records attributed to Jean-François Dat.

8 recordsLinked to original sources

arXiv2026

Parametrization and reduction to depth zero of $\overline{\mathbb{Z}}[\frac{1}{p}]$-blocks of tame $p$-adic groups

Let $G$ be a reductive group over a non-archimedean local field $F$ of residue characteristic $p$. We consider pairs $(ϕ,I)$ consisting of a "wild inertia" Langlands parameter $ϕ: P_F \longrightarrow \hat{G}$ whose centralizer $C_{\hat{G}}(ϕ)$ is a Levi subgroup of $\hat{G}$, and a cohomological invariant $I$ whose definition is inspired by the theory of endoscopy. Assuming that $p$ is odd and not a torsion prime of $G$ nor of $\hat{G}$, we associate to each such pair $(ϕ,I)$ a Serre subcategory $\mathrm{Rep}^{ϕ,I}(G(F))$ of the category of smooth $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of $G(F)$. Then we construct an equivalence between this Serre subcategory and the category of depth-zero $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of a twisted Levi subgroup $G_{ϕ,I}$ of $G$, which is dual to $C_{\hat{G}}(ϕ)$. This pattern for reduction to depth zero fits well with the conjectural (categorical) local Langlands correspondence. When $G$ is tamely ramified and $p$ does not divide the order of its Weyl group, then the above Serre subcategories provide the block decomposition of the category of all smooth $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of $G(F)$. In this case, we thus obtain a reduction-to-depth-zero process for smooth representations of $G(F)$ valued in any algebraically closed field of characteristic different from $p$. When that field has characteristic 0, this recovers some of the recent results of Adler--Fintzen--Mishra--Ohara. When that field is $\overline{\mathbb{F}}_{\ell}$, we use our results together with Zhu's unipotent categorical correspondence to produce a fully faithful embedding of $D\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\mathrm{GL}_{n}(F))$ into a suitable category of coherent sheaves on the moduli space of $n$-dimensional $\overline{\mathbb{F}}_{\ell}$-representations of the Weil group.

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arXiv2025

Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$

We consider the category of depth $0$ representations of a $p$-adic quasi-split reductive group with coefficients in $\overline{\mathbb{Z}}[\frac{1}{p}]$. We prove that the blocks of this category are in natural bijection with the connected components of the space of tamely ramified Langlands parameters for $G$ over $\overline{\mathbb{Z}}[\frac{1}{p}]$. As a particular case, this depth $0$ category is thus indecomposable when the group is tamely ramified. Along the way we prove a similar result for finite reductive groups. As an application, we deduce that the semi-simple local Langlands correspondence $π\mapsto φ_π$ constructed by Fargues and Scholze takes depth $0$ representations to tamely ramified parameters, using a motivic version of their construction recently announced by Scholze. We also bound the restriction of $φ_π$ to tame inertia in terms of the Deligne-Lusztig parameter of $π$ and show, in particular, that $φ_π$ is unramified if $π$ is unipotent.

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arXiv2024

Local Langlands in families: The banal case

We state a conjecture, local Langlands in families, connecting the centre of the category of smooth representations on $\mathbb{Z}[\sqrt{q}^{-1}]$-modules of a quasi-split $p$-adic group $\mathrm{G}$ (where $q$ is the cardinality of the residue field of the underlying local field), the ring of global functions on the stack of Langlands parameters for $\mathrm{G}$ over $\mathbb{Z}[\sqrt{q}^{-1}]$, and the endomorphisms of a Gelfand-Graev representation for $\mathrm{G}$. For a class of classical $p$-adic groups (symplectic, unitary, or split odd special orthogonal groups), we prove this conjecture after inverting an integer depending only on $\mathrm{G}$. Along the way, we show that the local Langlands correspondence for classical $p$-adic groups (1) preserves integrality of $\ell$-adic representations; (2) satisfies an "extended" (generic) packet conjecture; (3) is compatible with parabolic induction up to semisimplification (generalizing a result of Moussaoui), hence induces a semisimple local Langlands correspondence; and (4) the semisimple correspondence is compatible with automorphisms of $\mathbb{C}$ fixing $\sqrt{q}$.

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arXiv2024

Moduli of Langlands Parameters

Let $F$ be a nonarchimedean local field of residue characteristic $p$, let $\hat{G}$ be a split reductive group over $\mathbb{Z}[1/p]$ with an action of $W_F$, and let $^LG$ denote the semidirect product $\hat{G}\rtimes W_F$. We construct a moduli space of Langlands parameters $W_F \to {^LG}$, and show that it is locally of finite type and flat over $\mathbb{Z}[1/p]$, and that it is a reduced local complete intersection. We give parameterizations of the connected components and the irreducible components of the geometric fibers of this space, and parameterizations of the connected components of the total space over $\overline{\mathbb{Z}}[1/p]$ (under mild hypotheses) and over $\overline{\mathbb{Z}}_{\ell}$ for $\ell\neq p$. In each case, we show precisely how each connected component identifies with the "principal" connected component attached to a smaller split reductive group scheme. Finally we study the GIT quotient of this space by $\hat{G}$ and give a complete description of its fibers up to homeomorphism, and a complete description of its ring of functions after inverting an explicit finite set of primes depending only on $^LG$.

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arXiv2016

Derived categories and Deligne-Lusztig varieties II

This paper is a continuation and a completion of [BoRo1]. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups. We also show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of [BoRo1], the finer homotopy equivalence was unknown. As a consequence, the equivalence preserves defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations.

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arXiv2016

A functoriality principle for blocks of p-adic linear groups

Bernstein blocks of complex representations of p-adic reductive groups have been computed in a large amount of examples, in part thanks to the theory of types a la Bushnell and Kutzko. The output of these purely representation-theoretic computations is that many of these blocks are equivalent. The motto of this paper is that most of these coincidences are explained, and many more can be predicted, by a functoriality principle involving dual groups. We prove a precise statement for groups related to GL n , and then state conjectural generalizations in two directions : more general reductive groups and/or integral l-adic representations.

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arXiv2016

Equivalences of tame blocks for p-adic linear groups

Let p and $\ell$ be two distinct primes, F a p-adic field and n an integer. We show that any level 0 block of the category of smooth Z $\ell$-valued representations of GL n (F) is equivalent to the unipotent block of an appropriate product of GL n i (F i). To this end, we first show that the level 0 category of GL n (F) is equivalent to a category of " modules " over a certain Z $\ell$-algebra " with many objects " whose definition only involves n and the residue field of F. Then we use fine properties of Deligne-Lusztig cohomology to split this algebra and produce suitable Morita equivalences.

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arXiv2010

Théorie de Lubin-Tate non abélienne l-entière

For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a way compatible with extension of scalars. When R is a field of characteristic l, this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals, and a new proof of its existence. When R runs over complete local algebras, this provides bijections between deformations of matching mod-l representations. Roughly speaking, we can decompose "the supercuspidal part" of the l-integral cohomology as a direct sum, indexed by irreducible supercuspidals πmod l, of tensor products of universal deformations of πand of its two mates. Besides, we also get a virtual realization of both the semi-simple Langlands-Vigneras correspondence and the l-modular Langlands-Jacquet transfer for all representations, by using the cohomology complex and working in a suitable Grothendieck group.

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