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Vincent Sécherre

Publications and source records attributed to Vincent Sécherre.

At least 19 recordsLinked to original sources
arXiv2026

Godement-Jacquet gamma factors of distinguished representations of $\mathrm{GL}_n(\mathbb{F}_q)$

Let $k$ be a finite field of characteristic $p$. In the 1960s, Kondo attached non-abelian Gauss sums to irreducible $\mathbb{C}$-representations of $\mathrm{GL}_n(k)$, and computed them in terms of Green parameters. On the other hand, the Godement-Jacquet functional equation in which they occur was established by Macdonald in the 1980s. We first revisit Macdonald's and Kondo's results with a different perspective, in the process of generalizing their constructions to representations with coefficients in $\mathbb{Z}[\sqrt{p}^{-1},μ_p]$-algebras. Then, when $p$ is odd and $R$ is an algebraically closed field of characteristic different to $p$, our main result shows that the Godement-Jacquet gamma factor of a cuspidal irreducible $R$-representation, which is distinguished with respect to the subgroup fixed by a Galois or an inner involution, coincides with the sign of the associated period under the normalizer of this subgroup. Finally, we compute the gamma factors of these distinguished representations in terms of Green's and James' parametrizations of irreducible cuspidal $R$-representations.

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arXiv2026

Cuspidal $\ell$-modular representations of ${\rm GL}_n(F)$ distinguished by a Galois involution, II

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\neq2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of ${\rm GL}_n(F)$ which are ${\rm GL}_n(F_0)$-distinguished, that is, which carry a non-zero ${\rm GL}_n(F_0)$-invariant linear form. In the case when $\ell\neq2$, an $\ell$-modular cuspidal representation of ${\rm GL}_n(F)$ is ${\rm GL}_n(F_0)$-distinguished if and only if it lifts to a ${\rm GL}_n(F_0)$-distinguished cuspidal $\ell$-adic representation, whereas when $\ell=2$, it is ${\rm GL}_n(F_0)$-distinguished if and only if it is conjugate-self-dual.

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arXiv2026

Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods

For a non-Archimedean locally compact field $F$ of odd residue characteristic and characteristic $0$, we prove a conjecture of D. Prasad predicting that, for an integer $n \geq 1$ and a non-split quaternionic $F$-algebra $D$, a discrete series representation of ${\rm GL}_n(D)$ has a symplectic period if and only if it is cuspidal and its Jacquet--Langlands transfer to ${\rm GL}_{2n}(F)$ is non-cuspidal.

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arXiv2024

Local transfer for quasi-split classical groups and congruences mod l

Let G be the group of rational points of a quasi-split p-adic special orthogonal, symplectic or unitary group for some odd prime number p. FollowingArthur and Mok, there are a positive integer N, a p-adic field E and a local functorial transfer from isomorphism classes of irreducible smooth complex representations of G to those of GL(N,E). By fixing a prime number l different from p and an isomorphism between the field of complex numbers and an algebraic closure of the field of l-adic numbers, we obtain a transfer map between representations with l-adic coefficients. Now consider a cuspidal irreducible l-adic representation pi of G: we can define its reduction mod l, which is a semi-simple smooth representation of G of finite length, with coefficients in a field of characteristic l. Let pi' be a cuspidal irreducible l-adic representation of G whose reduction mod l is isomorphic to that of pi. We prove that the transfers of pi and pi' have reductions mod l which may not be isomorphic, but which have isomorphic supercuspidal supports. When G is not the split special orthogonal group SO(2), we further prove that the reductions mod l of the transfers of pi and pi' share a unique common generic component.

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arXiv2024

On modular rigidity for ${\rm GL}_n$

Let $k$ be a global field and $\mathbb{A}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to $\overline{\mathbb{Q}}_{\ell}$. Let $Π_1$ and $Π_2$ be cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A}_k)$ for some integer $n\geq1$. In this paper, we study the following question: assuming that there is a finite set $S$ of places of $k$ containing all Archimedean places and all finite places above $\ell$ such that, for all $v\notin S$, the local components $Π_{1,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ are unramified and their Satake parameters are congruent mod $\ell$, are the local components $Π_{1,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ integral, and do their reductions mod $\ell$ share an irreducible factor for all non-Archimedean places $w$ not dividing $\ell$? We show that, under certain conditions on $Π_1$ and $Π_2$, the answer is yes. We also give a simple proof when $k$ is a function field.

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arXiv2024

The sign of linear periods

Let $G$ be a group with subgroup $H$, and let $(π,V)$ be a complex representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$ on the space $\mathrm{Hom}_H(π,\mathbb{C})$ of $H$-invariant linear forms on $V$, provides a representation $χ_π$ of $N$ trivial on $H$, which is a character when $\mathrm{Hom}_H(π,\mathbb{C})$ is one dimensional. If moreover $G$ is a reductive group over a local field, and $π$ is smooth irreducible, it is an interesting problem to express $χ_π$ in terms of the possibly conjectural Langlands parameter $ϕ_π$ of $π$. In this paper we consider the following situation: $G=\mathrm{GL}_m(D)$ for $D$ a central division algebra of dimension $d^2$ over a local field $F$ of characteristic zero, $H$ is the centralizer of a non central element $δ\in G$ such that $δ^2$ is in the center of $G$, and $π$ has generic Jacquet-Langlands transfer to $\mathrm{GL}_{md}(F)$. In this setting the space $\mathrm{Hom}_H(π,\mathbb{C})$ is at most one dimensional. When $\mathrm{Hom}_H(π,\mathbb{C})\simeq \mathbb{C}$ and $H\neq N$, we prove that the value of the $χ_π$ on the non trivial class of $\frac{N}{H}$ is $(-1)^mε(ϕ_π)$ where $ε(ϕ_π)$ is the root number of $ϕ_π$. Along the way we extend many useful multiplicity one results for linear and Shalika models to the case of non split $G$. When $F$ is $p$-adic we also classify standard modules with linear periods and Shalika models, which are new results even when $D=F$.

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arXiv2023

Cuspidal $\ell$-modular representations of $\mathrm{GL}_n(F)$ distinguished by a Galois involution

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$ with Galois automorphism $σ$, and let $R$ be an algebraically closed field of characteristic $\ell\notin\{0,p\}$. We reduce the classification of $\mathrm{GL}_n(F_0)$-distinguished cuspidal $R$-representations of $\mathrm{GL}_n(F)$ to the level $0$ setting. Moreover, under a parity condition, we give necessary conditions for a $σ$-selfdual cuspidal $R$-representation to be distinguished. Finally, we classify the distinguished cuspidal $\overline{\mathbb{F}}_{\ell}$-representations of $\mathrm{GL}_n(F)$ having a distinguished cuspidal lift to $\overline{\mathbb{Q}}_\ell$.

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arXiv2023

Types et contragrédientes

Let G be a p-adic reductive group, and R an algebraically closed field. Let us consider a smooth representation of G on an R-vector space V. Fix an open compact subgroup K of G and a smooth irreducible representation of K on a finite-dimensional R-vector space W. The space of K-homomorphisms from W to V is a right module over the intertwining algebra H(G,K,W). We examine how those constructions behave when we pass to the contragredient representations of V and W, and we give conditions under which the behaviour is the same as in the case of complex representations. We take an abstract viewpoint and use only general properties of G. In the last section, we apply this to the theory of types for the group GL(n) and its inner forms over a non-Archimedean local field.

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arXiv2022

Représentations cuspidales de GL(r,D) distinguées par une involution intérieure

Let n be a positive integer, F be a non-Archimedean locally compact field of odd residue characteristic p and G be an inner form of GL(2n,F). This is a group of the form GL(r,D) for a positive integer r and division F-algebra D of reduced degree d such that rd=2n. Let K be a quadratic extension of F in the algebra of matrices of size r with coefficients in D, and H be its centralizer in G. We study selfdual cuspidal representations of G and their distinction by H from the point of view of type theory. Given such a representation pi of G, we compute the value at 1/2 of the epsilon factor of the restriction of the Langlands parameter phi of pi to the Weil group of K, denoted e(K,phi). When F has characteristic 0, we deduce that pi is H-distinguished if and only if phi is symplectic and e(K,phi)=(-1)^r. This proves in this case a conjecture by Prasad and Takloo-Bighash.

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arXiv2022

Block decomposition of the category of l-modular smooth representations of finite length of GL(m,D)

Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length smooth representations of G with coefficients in an algebraically closed field of characteristic l. Unlike the case of complex representations of an arbitrary p-adic reductive group and that of l-modular representations of GL(n,F), several non-isomorphic supercuspidal supports may correspond to the same block. We describe the (finitely many) supercuspidal supports corresponding to a given block. We also prove that a supercuspidal block is equivalent to the principal (that is, the one which contains the trivial character) block of the multiplicative group of a suitable division algebra, and we determine those irreducible representations having a nontrivial extension with a given supercuspidal representation of G.

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arXiv2021

The local Jacquet--Langlands correspondence and congruences modulo {\ell}

Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).

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arXiv2019

Supercuspidal representations of ${\rm GL}_n(F)$ distinguished by a Galois involution

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $σ$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $π$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $π^σ\simeqπ^{\vee}$ (such a representation is said to be $σ$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times}$. When $π$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $π$ to be ${\rm GL}_n(F_0)$-distinguished. When the characteristic $\ell$ of $R$ is not $2$, denoting by $ω$ the non-trivial $R$-character of $F_0^{\times}$ trivial on $F/F_0$-norms, we prove that any $σ$-selfdual supercuspidal $R$-representation is either distinguished or $ω$-distinguished, but not both. In the modular case, that is when $\ell>0$, we give examples of $σ$-selfdual cuspidal non-supercuspidal representations which are not distinguished nor $ω$-distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan-Kable-Tandon.

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arXiv2019

Galois self-dual cuspidal types and Asai local factors

Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $σ$ be its non-trivial automorphism. We show that any $σ$-self-dual cuspidal representation of ${\rm GL}_n(F)$ contains a $σ$-self-dual Bushnell--Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai $L$-function of a ${\rm GL}_n(F_{\mathsf{o}})$-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands--Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.

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arXiv2016

Towards an explicit local Jacquet-Langlands correspondence beyond the cuspidal case

We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let F be a non-Archimedean locally compact field of residue characteristic p, and let G be an inner form of the general linear group GL(n,F). We consider the problem of describing explicitly the local Jacquet--Langlands correspondence between the complex discrete series representations of G and GL(n,F), in terms of type theory. We show that the congruence properties of the local Jacquet--Langlands correspondence exhibited by A. Mínguez and the first named author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endo-class by the Jacquet--Langlands correspondence can be reduced to the case where the representations $π$ and its Jacquet--Langlands transfer JL($π$) are both cuspidal with torsion number 1. We also give an explicit description of the Jacquet--Langlands correspondence for all essentially tame discrete series representations of G, up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where $π$ and JL($π$) are both cuspidal.

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arXiv2015

Modular representations of GL(n) distinguished by GL(n-1) over a p-adic field

Let $\F$ be a non-Archimedean locally compact field, $q$ be the cardinality of its residue field, and $\R$ be an algebraically closed field of characteristic $\ell$ not dividing $q$.We classify all irredu\-cible smooth $\R$-representations of $\GL\_n(\F)$ having a nonzero $\GL\_{n-1}(\F)$-inva\-riant linear form, when $q$ is not congruent to $1$ mod $\ell$.Partial results in the case when $q$ is $1$ mod $\ell$ show that, unlike the complex case, the space of $\GL\_{n-1}(\F)$-invariant linear forms has dimension $2$ for certain irreducible representations.

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arXiv2015

Comptage de représentations cuspidales congruentes

Let $F$ be a non-Archimedean locally compact field of residue characteristic $p$, $G$ be an inner form of $GL_n(F)$, $n\ge1$, and $\ell$ be a prime number different from $p$. We give a numerical criterion for an integral $\ell$-adic irreducible cuspidal representation $\tildeρ$ of $G$ to have a super\-cuspidal irreducible reduction mod $\ell$, by counting inertial classes of cuspidal representations that are congruent to the inertial class of $\tildeρ$, generalizing results by Vign{é}ras and Dat. In the case the reduction mod $\ell$ of $\tildeρ$ is not super\-cuspidal irreducible, we show that this counting argument allows us to compute its length and the size of the supercuspidal support of its irreducible components. We define an invariant $w(\tildeρ)\ge1$ | the product of this length by this size | which is expected to behave nicely through the local Jacquet-Langlands correspondence. Given an $\ell$-modular irreducible cuspidal representation $ρ$ of $G$ and a positive integer $a$, we give a criterion for the existence of an integral $\ell$-adic irreducible cuspidal representation $\tildeρ$ of $G$ such that its reduction mod $\ell$ contains $ρ$ and has length $a$. This allows us to obtain a formula for the cardinality of the set of reductions mod $\ell$ of inertial classes of $\ell$-adic irreducible cuspidal representations $\tildeρ$ with given depth and invariant $w$. These results are expected to be useful to prove that the local Jacquet-Langlands correspondence preserves congruences mod $\ell$.

math.RT↗
arXiv2015

The $\ell$-modular Zelevinski involution

Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has characteristic $\ell>0$, the image of an irreducible smooth R-representation $π$ of G by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of G) contains a unique irreducible term $π$* with the same cuspidal support as $π$. This defines an involution on the set of isomorphism classes of irreducible R-representations of G, that coincides with the Zelevinski involution when R is the field of complex numbers. The method we use also works for F a finite field of characteristic p, in which case we get a similar result.

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