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Freydoon Shahidi

Publications and source records attributed to Freydoon Shahidi.

At least 19 recordsLinked to original sources
arXiv2026

On Jiang's wavefront sets conjecture for representations in local Arthur packets

This paper serves as an attempt towards the Jiang conjecture on the upper bound nilpotent orbits in the wavefront sets of representations in local Arthur packets of quasi-split classical groups, which is a natural generalization of the well-known Shahidi conjecture, reflecting the relation between the structure of wavefront sets and the local Arthur parameters. Applying the character identities of local Arthur packets and a matching method, we reduce the study of the upper bound to certain properties of the wavefront sets of the corresponding bi-torsor representations of general linear groups, which is implied by a recent result of Atobe and Ciubotaru for split classical groups when the residue characteristic is large.

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arXiv2025

Covering Barbasch-Vogan duality and wavefront sets of genuine representations

In this paper, we start by defining a covering Barbasch-Vogan duality and prove some of its properties. Then, for genuine representations of $p$-adic covering groups we formulate an upper bound conjecture for their wavefront sets using this covering Barbasch-Vogan duality and reduce it to anti-discrete representations. The formulation generalizes that of Ciubotaru-Kim and Hazeltine-Liu-Lo-Shahidi for linear algebraic groups. We prove this upper bound conjecture for Kazhdan-Patterson coverings of general linear groups.

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arXiv2025

$L$-packets and the generic Arthur packet conjectures for even unitary similitude groups

We establish the generic local Langlands correspondence by showing the equality of the Langlands-Shahidi $L$-functions and Artin $L$-functions in the case of even unitary similitude groups. As an application, we prove both weak and strong versions of the generic Arthur packet conjectures in the cases of even unitary similitude groups and even unitary groups. We further describe (not necessarily generic) $L$-packets for even unitary similitude groups and establish their expected properties, including Shahidi's conjecture, the finiteness of $L$-packets, and other related results.

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arXiv2025

The Jiang conjecture on the wavefront sets of local Arthur packets

This is a report on the progress made on a conjecture of Jiang on the upper bound nilpotent orbits in the wave front sets of representations in local Arthur packets of classical groups, which is a natural generalization of the Shahidi conjecture. We partially prove this conjecture, confirming the relation between the structure of wave front sets and the local Arthur parameters. Under certain assumptions, we also prove the enhanced Shahidi conjecture, which states that local Arthur packets are tempered if and only if they have generic members.

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arXiv2024

Rankin-Selberg L-Functions for GSpin x GL Groups

We construct an integral representation for the global Rankin-Selberg (partial) $L$-function $L(s, \pi \times \tau)$ where $\pi$ is an irreducible globally generic cuspidal automorphic representation of a general spin group (over an arbitrary number field) and $\tau$ is one of a general linear group, generalizing the works of Gelbart, Piatetski-Shapiro, Rallis, Ginzburg, Soudry and Kaplan among others. We consider all ranks and both even and odd general spin groups including the quasi-split forms. The resulting facts about the location of poles of $L(s, \pi \times \tau)$ have, in particular, important consequences in describing the image of the Langlands funtorial transfer from the general spin groups to general linear groups.

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arXiv2024

On anti-tempered local Arthur packets and a lemma of Arthur

In this paper, following Arthur's ideas, we rework the process of constructing the anti-tempered local Arthur packets for quasi-split classical groups and their pure inner forms. In particular, we present explicit examples illustrating certain gap in a consequential lemma of Arthur and provide a uniform modification, based on the work of Moeglin, Waldspurger, and Xu.

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arXiv2024

On the upper bound of wavefront sets of representations of p-adic groups

In this paper we study the upper bound of wavefront sets of irreducible admissible representations of connected reductive groups defined over non-Archimedean local fields of characteristic zero. We formulate a new conjecture on the upper bound and show that it can be reduced to that of anti-discrete series representations, namely, those whose Aubert-Zelevinsky duals are discrete series. Then, we show that this conjecture is equivalent to the Jiang conjecture on the upper bound of wavefront sets of representations in local Arthur packets and also equivalent to an analogous conjecture on the upper bound of wavefront sets of representations in local ABV packets.

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arXiv2021

On the Resolution of Reductive Monoids and Multiplicativity of $γ$-factors

In this article, we give a proof of multiplicativity for $γ$-factors, an equality of parabolically induced and inducing factors, in the context of the Braverman-Kazhdan/Ngo program, under the assumption of commutativity of the corresponding Fourier transforms and a certain generalized Harish-Chandra transform. We also discuss the resolution of singularities and their rationality for reductive monoids, which are among the basic objects in the program.

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arXiv2021

Restrictions, L-parameters, and local coefficients for genuine representations

We consider the restriction and induction of representations between a covering group and its derived subgroup, both on the representation-theoretic side and the L-parameter side. In particular, restriction of a genuine principal series is analyzed in detail. We also discuss a metaplectic tensor product construction for covers of the symplectic similitudes groups, and remark on the generality of such a construction for other groups. Furthermore, working with an arbitrary irreducible constituent of a unitary unramified principal series, we prove a multiplicity formula for its restriction to the derived subgroup in terms of three associated R-groups. Later in the paper, we study an unramified L-packet on how the parametrization of elements inside such a packet varies along with different choices of hyperspecial maximal compact subgroups and their splittings. We also investigate the genericity of elements inside such an L-packet with respect to varying Whittaker datum. Pertaining to the above two problems, covers of the symplectic similitudes groups are discussed in detail in the last part of the paper.

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arXiv2019

Local coefficients and gamma factors for principal series of covering groups

We consider an $n$-fold Brylinski-Deligne cover of a reductive group over a $p$-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman-Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of ${\rm GL}_2$ to ${\rm SL}_2$. In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.

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arXiv2018

On generalized Fourier Transforms for standard L-functions (with an appendix by Wen-Wei Li)

Any generalization of the method of Godement-Jacquet on principal L-functions for GL(n) to other groups as perceived by Braverman-Kazhdan and Ngo requires a Fourier transform on a space of Schwartz functions. In the case of standard L-functions for classical groups, a theory of this nature was developed by Piatetski-Shapiro and Rallis, called the doubling method. It was later that Braverman and Kazhdan, using an algebro-geometric approach, different from doubling method, introduced a space of Schwartz functions and a Fourier transform, which projected onto those from doubling method. In both methods a normalized intertwining operator played the role of the Fourier transform. The purpose of this paper is to show that the Fourier transform of Braverman-Kazhdan projects onto that of doubling method. In particular, we show that they preserve their corresponding basic functions. The normalizations involved are not the standard ones suggested by Langlands, but rather a singular version of local coefficients of Langlands-Shahidi method. The basic function will require a shift by 1/2 as dictated by doubling construction, reflecting the global theory, and begs explanation when compared with the work of Bouthier-Ngo-Sakellaridis. This matter is further discussed in an appendix by Wen-Wei Li.

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arXiv2017

Local Factors, Reciprocity and Vinberg Monoids

This article addresses the problem of existence of local factors, i.e., the root numbers and L-functions attached to representations of reductive groups over local fields and irreducible finite dimensional representations of their L-groups, as well as their equality with those of Artin factors through the local Langlands correspondence. We conclude the paper with a survey of the theory of monoids of Braverman-Kazhdan, Ngo and Vinberg in generalizing the method of Godement and Jacquet to arbitrary setting and their connections with Langlands-Shahidi method.

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arXiv2014

Local Transfer and Reducibility of Induced Representations of $p$-adic Groups of Classical Type

We analyze reducibility points of representations of $p$-adic groups of classical type, induced from generic supercuspidal representations of maximal Levi subgroups, both on and off the unitary axis. We are able to give general, uniform results in terms of local functorial transfers of the generic representations of the groups we consider. The existence of the local transfers follows from global generic transfers that were established earlier.

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arXiv2014

Local Langlands correspondence for $GL_n$ and the exterior and symmetric square $\varepsilon$--factors

Let $F$ be a $p$--adic field, i.e., a finite extension of $\mathbb Q_p$ for some prime $p$. The local Langlands correspondence attaches to each continuous $n$--dimensional $Φ$-semisimple representation $ρ$ of $W'_F$, the Weil--Deligne group for $\bar F/F$, an irreducible admissible representation $π(ρ)$ of $GL_n(F)$ such that, among other things, the local $L$- and $\varepsilon$-factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this paper, we show that this is the case for the local arithmetic and analytic symmetric square and exterior square $\varepsilon$--factors, that is, that $\varepsilon(s,Λ^2ρ,ψ)=\varepsilon(s,π(ρ),Λ^2,ψ)$ and $\varepsilon(s,Sym^2ρ,ψ)=\varepsilon(s,π(ρ),Sym^2,ψ)$. The agreement of the $L$-functions also follows by our methods, but this was already known by Henniart. The proof is a robust deformation argument, combined with local/global techniques, which reduces the problem to the stability of the analytic $γ$-factor $γ(s,π,Λ^2,ψ)$ under highly ramified twists when $π$ is supercuspidal. This last step is achieved by relating the $γ$-factor to a Mellin transform of a partial Bessel function attached to the representation and then analyzing the asymptotics of the partial Bessel function, inspired in part by the theory of Shalika germs for Bessel integrals. The stability for every irreducible admissible representation $π$ then follows from those of the corresponding arithmetic $γ$--factors as a corollary.

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arXiv2012

A p-adic integral for the reciprocal of L-functions

We introduce an analog of part of the Langlands-Shahidi method to the p-adic setting, constructing reciprocals of certain p-adic L-functions using the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group SL(2), and give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L-functions.

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arXiv2011

Functoriality for General Spin Groups

We establish the functorial transfer of generic, automorphic representations from the quasi-split general spin groups to general linear groups over arbitrary number fields, completing an earlier project. Our results are definitive and, in particular, we determine the image of this transfer completely and give a number of applications.

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arXiv2010

Arthur Packets and the Ramanujan Conjecture

The purpose of this paper is to show that under a part of generalized Arthur's A-packet conjecture, locally generic cuspidal automorphic representations of a quasisplit group over a number field are of Ramanujan type, i.e., are tempered at almost all places. The A-packet conjecture allows one to reduce the problem to a special case of a general local question about the components of the corresponding Langlands L-packet which is then answered here in its generality.

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