Jean-François Dat’s research while affiliated with Ecole Normale Supérieure de Paris and other places

Let F be a non-archimedean local field of residue characteristic p , let {\widehat G} be a split reductive group scheme over \mathbb{Z}[\frac{1}{p}] with an action of W_{F} , and let {}^{L}G denote the semidirect product \widehat{G}\rtimes W_{F} . We construct a moduli space of Langlands parameters W_{F} \rightarrow{}^{L}G , and show that it is locally of finite type and flat over \mathbb{Z}[\frac{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}}[\frac{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 \widehat{G} and give a 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 {}^{L}G .

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 $\sqrt{q}$.

Let G G be a reductive group over a non-archimedean local field F F of residue characteristic p p . We prove that the Hecke algebras of G ( F ) G(F) , with coefficients in any noetherian Z ℓ \mathbb {Z}_{\ell } -algebra R R with ℓ ≠ p \ell \neq p , are finitely generated modules over their centers, and that these centers are finitely generated R R -algebras. Following Bernstein’s original strategy, we then deduce that “second adjointness” holds for smooth representations of G ( F ) G(F) with coefficients in any Z [ 1 p ] \mathbb {Z}[\frac {1}{p}] -algebra. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain “excursion algebra” defined on the Langlands parameters side and the Bernstein center of G ( F ) G(F) . Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest.

Let G be a reductive group over a non-archimedean local field F of residue characteristic p. We prove that the Hecke algebras of G(F) with coefficients in a ${\mathbb Z}_{\ell}$-algebra R for $\ell$ not equal to p are finitely generated modules over their centers, and that these centers are finitely generated R-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of G(F) with coefficients in any ring R in which p is invertible. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain "excursion algebra" defined on the Langlands parameters side and the Bernstein center of G(F). Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest

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. We then outline a potential application to the Fargues-Scholze and Genestier-Lafforgue semisimple local Langlands correspondences. Namely, contingent on a certain "independence of $\ell$" property, our results imply that these correspondences take depth 0 representations to tamely ramified parameters.

Let F be a nonarchimedean local field of residue characteristic p, let $\hat{G}$ be a split connected 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 of this space over algebraically closed fields of characteristic zero and characteristic $\ell\neq p$, as well as of the components over $\overline{\mathbb{Z}}_{\ell}$ and (conjecturally) over $\overline{\mathbb{Z}}[1/p]$. Finally we study the functions on this space that are invariant under conjugation by $\hat{G}$ (or, equivalently, the GIT quotient by $\hat{G}$) and give a complete description of this ring of functions after inverting an explicit finite set of primes depending only on $^LG$.

This note is motivated by the problem of “uniqueness of supercuspidal support” in the modular representation theory of p-adic groups. We show that any counterexample to the same property for a finite reductive group lifts to a counterexample for the corresponding unramified p-adic group. To this end, we need to prove the following natural property: any simple subquotient of a parabolically induced representation is isomorphic to a subquotient of the parabolic induction of some simple subquotient of the original representation. The point is that we put no finiteness assumption on the original representation.

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.

Bernstein blocks of complex representations of p-adic reductive groups have been computed in a large number of examples, in part thanks to the theory of types à la Bushnell and Kutzko. The output of these purely representation-theoretic computations is that many of these blocks are equivalent. In this paper, we promote the idea 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 GLn, and then state conjectural generalizations in two directions: more general reductive groups and/or integral l-adic representations.

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.