arXiv subjects

Shaun Stevens

Publications and source records attributed to Shaun Stevens.

At least 19 recordsLinked to original sources

arXiv2026

The Local Langlands Correspondence for Middle Supercuspidal Representations of $p$-adic $\text{GL}(2n)$

Let $\text{F}$ be a non-archimedean local field of characteristic zero with residual characteristic $p$. In this paper we give an explicit description of the local Langlands correspondence for middle supercuspidal representations of $\text{GL}(2n,\text{F})$, under the tameness condition $p\nmid 2n$, in terms of the maximal simple types that define them. We achieve this by explicitly computing and comparing the gamma factors on the automorphic and Galois sides of the local Langlands correspondence. The computation on the automorphic side requires neither the tameness condition nor the characteristic zero condition.

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arXiv2026

Cuspidal endo-support and strong beta extensions

Let $G$ be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic $p$, assumed odd in the classical case. We prove that every smooth representation of $G$ over an algebraically closed field $R$ of characteristic $\ell\neq p$ contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to $G$, we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth $R$-representations of $G$.

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arXiv2025

On sharpness in Local Converse Theorems for classical groups and $G_2$

We prove various results about the Local Converse Problem for split reductive groups $G$ over a non-archimedean local field~$F$ of characteristic $0$ and residual characteristic $p$. In particular, we prove that when $G$ is a symplectic or special orthogonal group, or the exceptional group $G_2$, and $p$ is large enough, then the optimal standard Local Converse Theorem for $G(F)$ requires twisting by representations of $GL_r(F)$ with $r$ up to half the dimension of the standard representation of the dual group of $G$. However, if we restrict to generic supercuspidal representations of $G(F)$ then it can be improved when $G=SO_{2N}$; we conjecture that the same is true for symplectic and odd special orthogonal groups. We also consider the possibility of using non-standard representations of the dual group to distinguish representations, giving counterexamples to possible improvements for general linear groups, $G_2$ and $SO_{2N}$.

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arXiv2025

On the Local Converse Theorem for Depth $\frac{1}{N}$ Supercuspidal Representations of $\text{GL}(2N, F)$

In this paper, we use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $\text{GL}(2N, F)$ which we call middle supercuspidal representations. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, we show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times}$ and simple supercuspidal representations of $\text{GL}(N, F)$.

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arXiv2025

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

Reducibility points and characteristic $p$ local fields I- Simple supercuspidal representations of symplectic groups

Let $F$ be a non-Archimedean local field with odd characteristic $p$. Let $N$ be a positive integer and $G=Sp_{2N}(F)$. By work of Lomel\'i on $\gamma$-factors of pairs and converse theorems, a generic supercuspidal representation $\pi$ of $G$ has a transfer to a smooth irreducible representation $\Pi_\pi$ of $GL_{2N+1}(F)$. In turn the Weil-Deligne representation $\Sigma_\pi$ associated to $\Pi_\pi$ by the Langlands correspondence determines a Langlands parameter $\phi_\pi$ for $\pi$. That process produces a Langlands correspondence for generic cuspidal representations of $G$. In this paper we take $\pi$ to be simple in the sense of Gross and Reeder, and from the explicit construction of $\pi$ we describe $\Pi_\pi$ explicitly. The method we use is the same as in our previous paper arXiv:2310.20455, where we treated the case where $F$ is a $p$-adic field, and $\pi$ a simple supercuspidal representation of $G=Sp_{2N}(F)$. It relies on a criterion due to Moeglin on the reducibility of representations parabolically induced from $GL_M(F)\times G$ for varying positive integers $M$. We extend this criterion to the case when $F$ has any positive characteristic. The main new feature consists in relating reducibility to $\gamma$-factors for pairs.

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arXiv2024

Block decompositions for $p$-adic classical groups and their inner forms

For an inner form $\mathrm{G}$ of a general linear group or classical group over a non-archimedean local field of odd residue characteristic, we decompose the category of smooth representations on $\mathbb{Z}[\mu_{p^{\infty}},1/p]$-modules by endo-parameter. We prove that parabolic induction preserves these decompositions, and hence that it preserves endo-parameters. Moreover, we show that the decomposition by endo-parameter is the $\overline{\mathbb{Z}}[1/p]$-block decomposition; and, for $\mathrm{R}$ an integral domain, introduce a graph whose connected components parameterize the $\mathrm{R}$-blocks, in particular including the cases $\mathrm{R}=\overline{\mathbb{Z}}_{\ell}$ and $\mathrm{R}=\overline{\mathbb{F}}_\ell$ for $\ell\neq p$. From our description, we deduce that the $\overline{\mathbb{Z}_\ell}$-blocks and $\overline{\mathbb{F}_\ell}$-blocks of $\mathrm{G}$ are in natural bijection, as had long been expected. Our methods also apply to the trivial endo-parameter (i.e., the depth zero subcategory) of any connected reductive $p$-adic group, providing an alternative approach to results of Dat and Lanard in depth zero. Finally, under a technical assumption (known for inner forms of general linear groups) we reduce the $\mathrm{R}$-block decomposition of $\mathrm{G}$ to depth zero.

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arXiv2023

Simple cuspidal representations of symplectic groups: Langlands parameter

Let $F$ be a non-archimedean local field of odd residual characteristic. We compute the Jordan set of a simple cuspidal representation of a symplectic group over $F$, using explicit computations of generators of the Hecke algebras of covers reflecting the parabolic induction under study. When $F$ is a $p$-adic field we obtain the Langlands parameter of the representation.

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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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arXiv2017

Jordan blocks of cuspidal representations of symplectic groups

Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Moeglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a Ramification Theorem for $G$, giving a bijection between the set of endo-parameters for $G$ and the set of restrictions to wild inertia of discrete Langlands parameters for $G$, compatible with the local Langlands correspondence. The main tool consists in analysing the intertwining Hecke algebra of a good cover, in the sense of Bushnell--Kutzko, for parabolic induction from a cuspidal representation of $G\times\mathrm{GL}_n$, seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine its (ir)reducibility; a criterion of Moeglin then relates this to Langlands parameters.

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arXiv2016

On depth zero L-packets for classical groups

By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $\pi$ of a classical group (which may be not-quasi-split) over a nonarchimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing $\pi$. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to arbitrary tame Langlands parameters.

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arXiv2016

The regular representations of $\mathrm{GL}_{N}$ over finite local principal ideal rings

Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\geq2$ an integer. An irreducible representation of the finite group $G_{r}=\mathrm{GL}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+\mathfrak{p}^{r-1}\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ consists of representations whose stabilisers modulo $K^{1}$ are centralisers of regular elements in $\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_{r}$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_{r}$.

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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. M\'inguez 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 $\pi$ and its Jacquet--Langlands transfer JL($\pi$) 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 $\pi$ and JL($\pi$) are both cuspidal.

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arXiv2016

Intertwining semisimple characters for p-adic classical groups

Let~$G$ be a unitary group of an~$ε$-hermitian form~$h$ given over a nonarchimedean local field~$F_0$ of odd residue characteristic. We introduce a geometric combinatoric condition under which we prove "Intertwining implies Conjugacy" for semisimple characters of~$G$ and the general linear group of the ambient vector space of~$G$. Further we prove a Skolem-Noether result for the action of~$G$ on its Lie algebra, more precisely two Lie algebra elements of~$G$ which have the same characteristic polynomial over~$F$ must be conjugate under an element of~$G$ if there are corresponding semisimple characters which intertwine over an element of~$G$ } Let~$G$ be a unitary group over a nonarchimedean local field of odd residual characteristic. This paper concerns the study of the "wild part" of the irreducible smooth representations of~$G$, encoded in a so-called "semisimple character". We prove two fundamental results concerning them, which are crucial steps towards a classification of the cuspidal representations of~$G$. First we introduce a geometric combinatoric condition under which we prove an "intertwining implies conjugacy" theorem for semisimple characters, both in~$G$ and in the ambient general linear group. Second, we prove a Skolem--Noether theorem for the action of~$G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of~$G$ which have the same characteristic polynomial must be conjugate under an element of~$G$ if there are corresponding semisimple strata which are intertwined by an element of~$G$.

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arXiv2016

Endo-parameters for p-adic classical groups

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal C-representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart's notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

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arXiv2015

Cuspidal $\ell$-modular representations of $p$-adic classical groups

For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

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arXiv2015

Generalized Local Coefficients

In this paper we showed that under two assumptions we are able to define interesting functions that we call generalized local coefficients. We showed that in the quasi-split case generalized local coefficients are up to a positive constant the same as Shahidi's local coefficients. We provide a proof that the non quasi-split group $GL_m(D)$, for a central division algebra $D$ satisfies those assumptions. We also showed that generalized local coefficients satisfy nice properties, like the relation to Plancherel measures and multiplicativity inherited by that of intertwining operators. Generalized local coefficients are only defined for representations that are $(Y,φ)$-generic which is a generalization of generic representations in the quasi-split case. Here $Y$ denotes a nilpotent element in the Lie algebra of the group and $φ$ is a co-character related to $Y$.

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arXiv2015

Towards the Jacquet Conjecture on the Local Converse Problem for $p$-adic $\mathrm{GL}_n$

The Local Converse Problem is to determine how the family of the local gamma factors $γ(s,π\timesτ,ψ)$ characterizes the isomorphism class of an irreducible admissible generic representation $π$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $τ$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.

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