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Sarah Dijols

Publications and source records attributed to Sarah Dijols.

9 recordsLinked to original sources
arXiv2024

Whittaker normalization of $p$-adic ABV-packets and Vogan's conjecture for tempered representations

We show that ABV-packets for $p$-adic groups do not depend on the choice of a Whittaker datum, but the function from the ABV-packet to representations of the appropriate microlocal equivariant fundamental group does, and we find this dependence exactly. We study the relation between open parameters and tempered parameters and Arthur parameters and generic representations. We state a genericity conjecture for ABV-packets and prove this conjecture for quasi-split classical groups and their pure inner forms. Motivated by this we study ABV-packets for open parameters and prove that they are L-packets, and further that the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by the Langlands correspondence. From this conclude Vogan's conjecture on A-packets for tempered representations: ABV-packets for tempered parameters are Arthur packets and the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by Arthur.

math.RT↗
arXiv2024

Projection of root systems and the generalized injectivity conjecture for exceptional groups

Let $a$ be a real euclidean vector space of finite dimension and $Σ$ a root system in $a$ with a basis $Δ$. Let $Θ\subset Δ$ and $M = M_Θ$ be a standard Levi of a reductive group $G$ such that $a_Θ$ $= a_M / a_G$. Let us denote $d$ the dimension of $a_Θ$, i.e the cardinal of $Δ- Θ$ and $Σ_Θ$ the set of all non-trivial projections of roots in $Σ$. We obtain conditions on $Θ$ such that $Σ_Θ$ contains a root system of rank $d$. When considering the case of $Σ$ of type exceptional, we give a list of all exceptional root systems that can occur in $Σ_Θ$ and use it to prove the generalized injectivity conjecture in most exceptional groups cases.

math.RT↗
arXiv2024

Projection of root systems

Let $a$ be a real euclidean vector space of finite dimension and $Σ$ a root system in $a$ with a basis $Δ$. Let $Θ\subset Δ$ and $M = M_Θ$ be a standard Levi of a reductive group $G$ such that $a_Θ= a_M / a_G$. Let us denote $d$ the dimension of $a_Θ$, i.e the cardinal of $Δ- Θ$ and $Σ_Θ$ the set of all non-trivial projections of roots in $Σ$. We obtain conditions on $Θ$ such that $Σ_Θ$ contains a root system of rank $d$.

math.RT↗
arXiv2024

Generic representations, open parameters and ABV-packets for $p$-adic groups

If $π$ is a representation of a $p$-adic group $G(F)$, and $ϕ$ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of $ϕ$ that will detect when $π$ is generic? In this paper we show that if $G$ is classical or if we assume the Kazhdan-Lusztig hypothesis for $G$, then the answer is yes, and the property is that the orbit of $ϕ$ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter $ϕ$ for a $p$-adic group $G(F)$, the ABV-packet $Π^{\mathrm{ABV}}_ϕ(G(F))$ contains a generic representation if and only if the local adjoint L-function $L(s,ϕ,\mathop{\text{Ad}})$ is regular at $s=1$, and show that this condition is equivalent to the "open parameter" condition above. We show that this genericity conjecture for ABV-packets follows from other standard conjectures and we verify its validity with the same conditions on $G$. We show that, in this case, the ABV-packet for $ϕ$ coincides with its $L$-packet. Finally, we prove Vogan's conjecture on $A$-packets for tempered parameters.

math.RT↗
arXiv2023

Period-index in top cohomology over semiglobal fields

We prove a common slot lemma for symbols in top cohomology classes over semiglobal fields. Furthermore, we prove that period and index agree for general top cohomology classes over such fields. We discuss applications to quadratic forms and related open problems.

math.NT↗
arXiv2023

Parabolically induced representations of p-adic $G_2$ distinguished by $SO_4$, I

We consider the parabolically induced representations of the symmetric space $SO_4\backslash G_2$ over a p-adic field using the geometric lemma when the inducing parabolic is $P_β$. Using an explicit description of the embedding of $G_2$ in $GL_8$, we characterize precisely the induced representations which are $(SO_4, χ)$-distinguished, given a certain type of involutions is chosen.

math.RT↗
arXiv2022

The completed standard $L$-function of modular forms on $G_2$

The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(π,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $π$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $π$ of $G_2(\mathbb{A})$ that corresponds to a modular form $φ$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $Λ(π,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $φ$ is nonzero, we prove the functional equation $Λ(π,\operatorname{Std},s) = Λ(π,\operatorname{Std},1-s)$. Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal.

math.NT↗
arXiv2022

The Generalized Injectivity Conjecture

We prove a conjecture of Casselman and Shahidi stating that the unique irreducible generic subquotient of a standard module is necessarily a subrepresentation for a large class of connected quasi-split reductive groups, in particular for those which have a root system of classical type (or product of such groups). To do so, we prove and use the existence of strategic embeddings for irreducible generic discrete series representations, extending some results of Moeglin.

math.NT↗
arXiv2018

Symplectic models for Unitary groups

In analogy with the study of representations of $GL_{2n}(F)$ distinguished by $Sp_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$. (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain $Sp_{2n}(F)$.) We prove that there are no cuspidal representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$ for $F$ a non-archimedean local field. We also prove the corresponding global theorem that there are no cuspidal representations of $U_{2n}({\mathbb A}_k)$ with nonzero period integral on $Sp_{2n}(k) \backslash Sp_{2n}({\mathbb A}_k)$ for $k$ any number field or a function field. We completely classify representations of quasi-split unitary group in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasi-split unitary group distinguished by $Sp_{2n}(F)$.

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