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Marie-France Vignéras

Publications and source records attributed to Marie-France Vignéras.

9 recordsLinked to original sources
arXiv2025

Représentations des quaternions de norme 1

Let F be a local field with finite residue field of characteristic p, D the quaternion division algebra with centre F, and R an algebraically closed field of any characteristic. We classify the smooth irreducible R-representations V of the group D^1 of elements of D* with reduced norm 1. Such a V occurs in the restriction of a smooth irreducible R-representation V* of D*. When the dimension of V*is >1, following our previous work in the case of SL_2(F), we show that the restriction of V* to D^1 is irreducible or the sum of two irreducible representations. When the characteristic of R is not p, that restriction is the sum of two irreducible equivalent representations if and only if the representation of GL_2(F) image of V* by the Jacquet-Langlands correspondence restricts to SL_2(F) as a sum of four inequivalent irreducible representations (this is never the case if the characteristic of R is not 2).

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arXiv2025

Representations of $SL_2(F)$

Let $p$ be a prime number, $F $ a non-archimedean local field with residue characteristic $p$, and $R$ an algebraically closed field of characteristic different from $ p$. We thoroughly investigate the irreducible smooth $R$-representations of $SL_2(F)$. The components of an irreducible smooth $R$-representation $Π$ of $GL_2(F)$ restricted to $SL_2(F)$ form an $L$-packet $L(Π)$. We use the classification of such $Π$ to determine the cardinality of $L(Π)$, which is $1,2$ or $4$. When $p=2$ we have to use the Langlands correspondence for $GL_2(F)$. When $\ell$ is a prime number distinct from $p$ and $R=\mathbb Q_\ell^{ac}$, we establish the behaviour of an integral $L$-packet under reduction modulo $\ell$. We prove a Langlands correspondence for $SL_2(F)$, and even an enhanced one when the characteristic of $R$ is not $2$. Finally, pursuing a theme of \cite{HV23}, which studied the case of inner forms of $GL_n(F)$, we show that near identity an irreducible smooth R-representation of $SL_2(F)$ is, up to a finite dimensional representation, isomorphic to a sum of $1,2$ or $4$ representations in an $L$-packet of size $4$ (when $p$ is odd there is only one such $L$-packet).

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arXiv2022

Representations of $p$-adic groups over coefficient rings

Motivated by the Langlands program in representation theory, number theory and geometry, the theory of representations of a reductive $p$-adic group over a coefficient ring different from the field of complex numbers has been widely developped during the last two decades. This article provides a survey of basic results obtained in the 21st century.

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arXiv2022

Inverse Satake isomorphism and change of weight

Let $G$ be any connected reductive $p$-adic group. Let $K\subset G$ be any special parahoric subgroup and $V,V'$ be any two irreducible smooth $\overline {\mathbb F}_p[K]$-modules. The main goal of this article is to compute the image of the Hecke bi-module $\operatorname{End}_{\overline {\mathbb F}_p[K]}(\operatorname{c-Ind}_K^G V, \operatorname{c-Ind}_K^G V')$ by the generalized Satake transform and to give an explicit formula for its inverse, using the pro-$p$ Iwahori Hecke algebra of $G$. This immediately implies the "change of weight theorem" in the proof of the classification of mod $p$ irreducible admissible representations of $G$ in terms of supersingular ones. A simpler proof of the change of weight theorem, not using the pro-$p$ Iwahori Hecke algebra or the Lusztig-Kato formula, is given when $G$ is split (and in the appendix when $G$ is quasi-split, for almost all $K$).

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arXiv2021

Representations of a reductive $p$-adic group in characteristic distinct from $p$

We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over a field $C$ of characteristic different from $p$. When $C$ is algebraically closed, for many groups $G$, a list of cuspidal $C$-types $(J,λ)$ has been produced satisfying exhaustion, sometimes for a restricted kind of cuspidal representations, and often unicity. We verify that those lists verify Aut($C$)-stability and we produce similar lists when $C$ is no longer assumed algebraically closed. Our other main results concern supercuspidality. This notion makes sense for the representations $λ$ in the cuspidal $C$-types $(J,λ)$ as above, which involve finite reductive groups. We check that an irreducible cuspidal representation of $G$ induced from $λ$ is supercuspidal if and only $λ$ is supercuspidal.

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arXiv2017

Existence of supersingular representations of p-adic reductive groups

Let G be a connected simple adjoint p-adic group not isomorphic to a projective linear group PGL(m,D) of a division algebra D, or an adjoint ramified unitary group of a split hermitian form in 3 variables. We prove that G admits an irreducible admissible supercuspidal (i.e. supersingular) representation over any field of characteristic p.

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arXiv2017

Modulo $p$ representations of reductive $p$-adic groups: functorial properties

Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $τ$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G τ$ and we show that $\mathrm{Ind}_P^G χτ$ is irreducible for a general unramified character $χ$ of $M$. In the reverse direction, we compute the image by the two adjoints of $\mathrm{Ind}_P^G$ of an irreducible admissible representation $π$ of $G$. On the way, we prove that the right adjoint of $\mathrm{Ind}_P^G $ respects admissibility, hence coincides with Emerton's ordinary part functor $\mathrm{Ord}_{\overline{P}}^G$ on admissible representations.

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