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Elad Zelingher

Publications and source records attributed to Elad Zelingher.

16 recordsLinked to original sources
arXiv2026

Kloosterman sheaves and Bessel functions for generic principal series of finite groups

Let $G$ be a quasi-split reductive group over a finite field and let $\^G$ be the Langlands dual group. Assuming the derived subgroup of $G$ is almost simple, we use techniques from the geometric Langlands program to relate special values of Bessel functions for generic principal series representations of $G$ to the trace of Frobenius acting on Kloosterman sheaves of Heinloth-Ng\^o-Yun for $\^G$. We also give explicit examples in essentially all possible cases, including exceptional groups.

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arXiv2026

On the Godement-Jacquet functional equation for finite matrix monoids

In previous work with Harman and Snowden, we found explicit formulas for units of rank ideals of the matrix monoid algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where $\mathbb{K}$ is an algebraically closed field with characteristic not dividing $q$. In this work, we use these explicit formulas to establish a regularized Godement--Jacquet functional equation valid for irreducible representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and of the algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$. Using this functional equation we give an explicit expression for primitive central idempotents corresponding to irreducible representations of $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where the characteristic of $\mathbb{K}$ does not divide $\left|\operatorname{GL}_n\left(\mathbb{F}_q\right)\right|$. Interestingly, the coefficients in this formula are closely related to degenerate non-abelian Gauss sums.

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arXiv2026

On classical doubling method gamma factors for certain depth zero representations

Piatetski-Shapiro--Rallis discovered an integral representation construction, known as the doubling method, for the tensor product $L$-function of a cuspidal automorphic representation of $G \times \mathrm{GL}_1$, where $G$ is a classical group. Lapid--Rallis defined and studied the counterpart local factors. In this article, following Lapid--Rallis, we define and study an analogous doubling method gamma factor associated to irreducible representations of classical finite groups of Lie type. We prove that this gamma factor is multiplicative and use results of Yost-Wolff--Zelingher to give explicit formulas for it in terms of the Deligne--Lusztig data of the representation in the non-conjugate-dual character case. Finally, we relate our construction to the local construction of Lapid--Rallis via certain depth zero supercuspidal representations of classical groups.

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arXiv2025

On Jacobi sums arising from the classical doubling method

We define the notion of a non-abelian Jacobi sum $\mathcal{J}^{\mathrm{dbl}}\left(\pi, \chi\right)$ attached to an irreducible representation $\pi$ of a general linear group or a classical group over a finite field and a character $\chi$ of the multiplicative group of the finite field or its quadratic extension. These sums emerge in the study of the doubling method of Piatetski-Shapiro--Rallis and Lapid--Rallis. For general linear groups, we express these non-abelian Jacobi sums in terms of Kondo's non-abelian Gauss sums. For classical groups and for characters that are not conjugate-dual, we give an explicit formula for these non-abelian Jacobi sums in terms of Gauss sums attached to the Deligne--Lusztig data of the representation, and we prove that these Jacobi sums are constant on geometric Lusztig series. Our results rely on a multiplicativity result of non-abelian Jacobi sums obtained by Girsch--Zelingher.

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arXiv2025

Representations of finite matrix monoids

Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem that the two-sided ideal spanned by matrices of rank at most $r$ has a unit. Our most significant result is an explicit formula for this unit. Prior to our work, such a formula was only known in a few examples. We also study the module theory of $\mathbf{C}[\mathfrak{M}_n]$. We explicitly describe the simple modules, and establish induction and restriction rules. We show that the simple decomposition of an arbitrary module can be determined using character theory of finite general linear groups; this relies on a Pieri rule of Gurevich--Howe. We also establish a version of Schur--Weyl duality for $\mathbf{C}[\mathfrak{M}_n]$. Many of these results hold over more general coefficient fields.

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arXiv2025

On exotic matrix exponential sums and Bessel-Speh functions

In a previous work with Carmon, we defined Bessel--Speh functions. These are matrix coefficients of irreducible Speh representations of $\mathrm{GL}_{kc}(\mathbb{F})$, where $\mathbb{F}$ is a finite field. They arise from $(k,c)$ models, which are models that generalize the Whittaker model to Speh representations attached to irreducible generic representations. These constructions are finite field analogs of objects arising naturally in the generalized doubling method over $p$-adic fields, a recently active area of the Langlands program. In this article we study special values of Bessel--Speh functions which were used in our previous work with Carmon to define Ginzburg--Kaplan gamma factors. Our main result computes the special values of interest explicitly in terms of new arithmetic objects we introduce, called exotic matrix Kloosterman sums, which generalize both Katz's exotic Kloosterman sums and twisted matrix Kloosterman sums. We then show that exotic matrix Kloosterman sums can be expressed as products of modified Hall--Littlewood polynomials evaluated at roots of the characteristic polynomial of the Frobenius acting on Katz's exotic Kloosterman sheaf. As an application of our results, we establish new identities for Bessel functions of irreducible generic representations.

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arXiv2025

On tori periods of Weil representations of unitary groups

We determine the restriction of Weil representations of unitary groups to maximal tori. In the local case, we show that the Weil representation contains a pair of compatible characters if and only if a root number condition holds. In the global case, we show that a torus period corresponding to a maximal anisotropic torus of the global theta lift of a character does not vanish if and only if the local condition is satisfied everywhere and a central value of an $L$-function does not vanish. Our proof makes use of the seesaw argument and of the well-known theta lifting results from $\operatorname{U}\left(1\right)$ to $\operatorname{U}\left(1\right)$. Our results are used in other papers to construct Arthur packets for $G_2$.

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arXiv2025

On a Casselman-Shalika type formula for unramified Speh representations

We give a Casselman-Shalika type formula for unramified Speh representations. Our formula computes values of the normalized spherical element of the $(k,c)$ model of a Speh representation at elements of the form $\operatorname{diag}\left(g, I_{(k-1)c}\right)$, where $g \in \mathrm{GL}_c\left(F\right)$ for a non-archimedean local field $F$. The formula expresses these values in terms of modified Hall--Littlewood polynomials evaluated at the Satake parameter attached to the representation. Our proof is combinatorial and very simple. It utilizes Macdonald's formula and the unramified computation of the Ginzburg--Kaplan integral. This addresses a question of Lapid-Mao.

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arXiv2025

On matrix Kloosterman sums and Hall-Littlewood polynomials

We prove an identity relating twisted matrix Kloosterman sums to modified Hall-Littlewood polynomials evaluated at the roots of the characteristic polynomial associated to a twisted Kloosterman sheaf. This solves a conjecture of Erdélyi and Tóth.

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arXiv2024

On Speh representations for level zero supercuspidal representations and Ginzburg-Kaplan gamma factors

We establish a relation between Speh representations of $\mathrm{GL}_n\left(\mathbb{F}_q\right)$ and Speh representations of $\mathrm{GL}_n\left(F\right)$, where $F$ is a non-archimedean local field. We use irreducible level zero supercuspidal representations to show that these two notions of Speh representations associated to cuspidal representations are related via a commutative diagram, and that their corresponding $(k,c)$ $ψ$-Whittaker models are also related. We use these results to relate the local Ginzburg-Kaplan integrals for level zero supercuspidal representations to their finite field counterparts.

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arXiv2024

On Ginzburg-Kaplan gamma factors and Bessel-Speh functions for finite general linear groups

We give a new construction of tensor product gamma factors for a pair of irreducible representations of $\operatorname{GL}_c\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_k\left(\mathbb{F}_q\right)$. This construction is a finite field analog of a construction of doubling type due to Kaplan in the local field case and due to Ginzburg in the global case, and it only assumes that one of the representations in question is generic. We use this construction to establish a relation between special values of Bessel functions attached to Speh representations of generic principal series representations and twisted matrix Kloosterman sums. Using this relation, we establish the multiplicativity identity of twisted matrix Kloosterman sums.

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arXiv2024

On values of the Bessel function for generic representations of finite general linear groups

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values of the Bessel function can be realized as the coefficients of $L$-functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.

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arXiv2024

On gamma factors for representations of finite general linear groups

We use the Langlands--Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_m\left(\mathbb{F}_q\right)$. We prove that the Shahidi gamma factor is multiplicative and show that it is related to the Jacquet--Piatetski-Shapiro--Shalika gamma factor. As an application, we prove a converse theorem based on the absolute value of the Shahidi gamma factor, and improve the converse theorem of Nien. As another application, we give explicit formulas for special values of the Bessel function of an irreducible generic representation of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$.

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arXiv2020

Epsilon factors of representations of finite general linear groups

We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums. The exterior square epsilon factors relate the Jacquet-Shalika exterior square gamma factors and the Langlands-Shahidi exterior square gamma factors for level zero supercuspidal representations. We prove that these exterior square factors coincide in a special case.

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arXiv2020

Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: finite field analogs and level zero representations

We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exterior square gamma factors for irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors are expressed in terms of Bessel functions, or in terms of the regular characters associated with the cuspidal representations. We also relate our exterior square gamma factors over finite fields to those over local fields through level zero representations.

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