arXiv subjects

Justin Trias

Publications and source records attributed to Justin Trias.

9 recordsLinked to original sources

arXiv2026

Modular Weil representation and compatibility of cuspidals with congruences

Let $F$ be a non-archimedean local field of characteristic different from $2$ and of residual characteristic $p$. We generalise the theory of the Weil representation over $F$ with complex coefficients to $\ell$-modular representations \textit{i.e.} when the complex coefficients are replaced by a coefficient field $R$ of characteristic $\ell \neq p$. We obtain along the way a generalisation of the Stone-von Neumann theorem to the $\ell$-modular setting, together with the Weil representation with coefficients in $R$ on the $R$-metaplectic group. Surprisingly enough, the latter $R$-metaplectic group happens to be split over the symplectic group if $\ell = 2$. The theory also makes sense when $F$ is a finite field of odd characteristic. We also establish the irreducibility of the theta lift in the cuspidal case as long as $\ell$ does not divide the pro-orders of the groups at stake and we provide a compatibility to congruences in this setting via an integral version of the theta lift.

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arXiv2026

On the rationality of the Weil Representation and the local theta correspondence

We prove that the Weil representation over a non-archimedean local field can be realised with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\ell$-modular coefficient fields, as well as coefficient rings such as rings of integers i.e. in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational.

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arXiv2025

The MVW involution of the metaplectic group

The MVW involution -- named after Colette Moeglin, Marie-France Vign\'eras, and Jean-Loup Waldspurger -- is a fundamental dualizing involution in the representation theory of $p$-adic classical groups. It extends the well-known transpose-inverse automorphism for general linear groups. In this work, we establish the existence of the MVW involution for the metaplectic group over a non-archimedean local field $F$ of characteristic different from $2$ and with residue characteristic $p$. Our construction applies to representations over any coefficient field of characteristic distinct from $p$.

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arXiv2025

The universal Harish-Chandra $j$-function

Let $F$ be a nonarchimedean local field with residue field of cardinality $q$, let $G$ be the $F$-points of a connected reductive group defined over $F$, let $P$ and $Q$ be two parabolic subgroups with the same Levi factor $M$. We construct intertwining operators $J_{Q|P}$ and the Harish-Chandra $j$-function $j^G$ for finitely generated smooth $A[M]$-modules, where $A$ is any commutative Noetherian algebra over $\mathbb{Z}' :=\mathbb{Z}[\sqrt{q}^{-1}]$. The construction is functorial, compatible with extension of scalars, and generalizes the previously known constructions. We prove a generic Schur's lemma result for parabolic induction, which circumvents the need for generic irreducibility in defining $j^G$. Setting $A=\mathbb{Z}'$ and applying the construction to finitely generated projective generators produces a universal $j$-function that is a rational function with coefficients in the Bernstein center of $M$ over $\mathbb{Z}'$, and which gives the $j$-function of any object via specializing at points of the Bernstein scheme. We conclude by characterizing the local Langlands in families morphism (when it exists) for quasisplit classical groups in terms of an equality of $j$-functions.

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arXiv2025

The $\ell$-modular local theta correspondence

We study the validity of the local theta correspondence over a non-archimedean local field in the context of modular representation theory \textit{i.e.} for representations with coefficient fields of positive characteristic. For a symplectic-orthogonal or a unitary-unitary dual pair over a $p$-adic field, we obtain a bijective correspondence, as long as the characteristic of the coefficient field is large enough compared to the size of the dual pair, and call it the modular local theta correspondence.

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arXiv2023

Towards a theta correspondence in families for type II dual pairs

Let $R$ be a commutative $\mathbb{Z}[1/p]$-algebra, let $m \leq n$ be positive integers, and let $G_n=\text{GL}_n(F)$ and $G_m=\text{GL}_m(F)$ where $F$ is a $p$-adic field. The Weil representation is the smooth $R[G_n\times G_m]$-module $C_c^{\infty}(\text{Mat}_{n\times m}(F),R)$ with the action induced by matrix multiplication. When $R=\mathbb{C}$ or is any algebraically closed field of banal characteristic compared to $G_n$ and $G_m$, the local theta correspondence holds by the work of Howe and M\'inguez. At the level of supercuspidal support, we interpret the theta correspondence as a morphism of varieties $\theta_R$, which we describe as an explicit closed immersion. For arbitrary $R$, we construct a canonical ring homomorphism $\theta^\#_{R} : \mathfrak{Z}_{R}(G_n)\to \mathfrak{Z}_{R}(G_m)$ that controls the action of the center $\mathfrak{Z}_{R}(G_n)$ of the category of smooth $R[G_n]$-modules on the Weil representation. We use the rank filtration of the Weil representation to first obtain $\theta_{\mathbb{Z}[1/p]}^\#$, then obtain $\theta^\#_R$ for arbitrary $R$ by proving $\mathfrak{Z}_R(G_n)$ is compatible with scalar extension. In particular, the map $\text{Spec}(\mathfrak{Z}_R(G_m))\to \text{Spec}(\mathfrak{Z}_R(G_n))$ induced by $\theta_R^\#$ recovers $\theta_R$ in the $R=\mathbb{C}$ case and in the banal case. We use gamma factors to prove $\theta_R^\#$ is surjective for any $R$. Finally, we describe $\theta^\#_R$ in terms of the moduli space of Langlands parameters and use this description to give an alternative proof of surjectivity in the tamely ramified case.

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arXiv2022

Whittaker functionals and contragredient in characteristic not $p$

Let $R$ be an algebraically closed field and $\ell$ be its characteristic. Let $G$ be a locally profinite group having a compact open subgroup of invertible pro-order in $R$. Take $N$ a closed subgroup of $G$ exhausted by compact subgroups of invertible pro-orders in $R$ and fix a smooth character $\theta$ of $N$. For $\pi$ an irreducible smooth $R$-representation of $G$ whose matrix coefficients are compactly supported modulo the center (we call it $Z$-compact), we show that the dimensions $\mathrm{Hom}_{N}(\pi,\theta)$ and $\mathrm{Hom}_{N}(\pi^\vee,\theta^{-1})$ are equal provided one of the two is finite. We derive a few applications from this result. First, we prove that any $G$-intertwiner from $\pi$ to $\mathrm{Ind}_N^G(\theta)$ has image in $\mathrm{ind}_{ZN}^G(\omega_\pi \theta)$, where $\omega_\pi$ is the central character of $\pi$, and the Whittaker space of $\pi$ agrees with that of its Whittaker periods. Second, it applies to quasi-split groups over non Archimedean local fields of residual characteristic $p \neq \ell$ and where $N$ is the unipotent radical of a Borel subgroup of $G$ together with a generic character $\theta$. Our equality of dimensions turns out to be a good replacement for Rodier's crucial use of complex conjugation in the proof of Whittaker multiplicity at most one for cuspidal representations. Then by a lifting argument, we recover Rodier's generalization of the Gelfand-Kazhdan property for $R$-valued $(\theta^{-1}\otimes \theta)$-equivariant distributions on $G$. This latter fact, together with Rodier's heridity property, which is valid in our context, leads to the multiplicity at most one of Whittaker functionals over $R$. We also give other applications, including a generalization over $R$ of a result for complex representations proved by Chang Yang and initially conjectured by Dipendra Prasad.

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arXiv2021

Universal Weil module

The classical construction of the Weil representation, with complex coefficients, has long been expected to work for more general coefficient rings. This paper exhibits the minimal ring $\mathcal{A}$ for which this is possible, the integral closure of $\mathbb{Z}[\frac{1}{p}]$ in a cyclotomic field, and carries out the construction of the Weil representation over $\mathcal{A}$-algebras. As a leitmotif all along the work, most of the problems can actually be solved over the base ring $\mathcal{A}$ and transferred to any $\mathcal{A}$-algebra by scalar extension. The most striking fact is that all these Weil representations arise as the scalar extension of a single one with coefficients in $\mathcal{A}$. In this sense, the Weil module obtained is universal. Building upon this construction, we speculate and make predictions about an integral theta correspondence.

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arXiv2020

Correspondance th\^eta locale $\ell$-modulaire I : groupe m\'etaplectique, repr\'esentation de Weil et $\Theta$-lift

Let $F$ be a field which is, either local non archimedean, or finite, of residual charcateristic $p$ but of characteristic different from $2$. Let $W$ be a symplectic space of finite dimension over $F$. Suppose $R$ is a field of characteristic $\ell \neq p$ so that there exists a non trivial smooth additive character $\psi : F \to R^\times$. Then the Stone-von Neumann theorem of the Heisenberg group $H(W)$ is still valid for representations with coefficients in $R$. It leads to a projective representation of the group $\text{Sp}(W)$ which lifts to a genuine smooth representation of a central extension of $\text{Sp}(W)$ by $R^\times$: this is the modular Weil representation of the metaplectic group. For any dual pair $(H_1,H_2)$, their lifts to the metaplectic group may splitor not according to the different cases at stake. Eventually, computing the biggest isotypic quotient of the modular Weil representation allows to define the $\Theta$-lift. Some new lines of investigation are thus available with these new tools such as studying scalar extension and reduction modulo $\ell$.

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