arXiv subjects

Eyal Kaplan

Publications and source records attributed to Eyal Kaplan.

28 records · Page 2Linked to original sources

arXiv2019

On the Langlands parameter of a simple supercuspidal representation: even orthogonal groups

Let $π$ be a simple supercuspidal representation of the split even special orthogonal group. We compute the Rankin-Selberg $γ$-factors for rank 1-twists of $π$ by quadratic tamely ramified characters of $F^*$. We then use our results to determine the Langlands parameter of $π$ up to its restriction to the wild inertia subgroup, subject to an analogue of a work of Blondel, Henniart, and Stevens for $SO_{2l}$. In the particular case of the field $\mathbb{Q}_2$, we are able to describe the parameter completely.

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arXiv2018

Doubling Constructions and Tensor Product ${L}$-Functions: the linear case

We present an integral representation for the tensor product $L$-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. This is the first in a series of papers, treating symplectic and even orthogonal groups. Subsequent papers (in preparation) will treat odd orthogonal and general spin groups, the metaplectic covering version of these integrals, and applications to functoriality coming from combining this work with the converse theorem (and independent of the trace formula).

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arXiv2018

The Langlands parameter of a simple supercuspidal representation: Symplectic groups

Let $π$ be a simple supercuspidal representation of the symplectic group $Sp_{2l}(F)$, over a $p$-adic field $F$. In this work, we explicitly compute the Rankin-Selberg $γ$-factor of rank-$1$ twists of $π$. We then completely determine the Langlands parameter of $π$, if $p \neq 2$. In the case that $F = \mathbb{Q}_2$, we give a conjectural description of the functorial lift of $π$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.

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arXiv2016

A Godement-Jacquet type integral and the metaplectic Shalika model

We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.

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arXiv2016

Doubling Constructions for Covering Groups and Tensor Product L-Functions

This is a research announcement concerning a series of constructions obtained by applying the "doubling method" from the theory of automorphic forms to covering groups. Using these constructions, we obtain partial tensor product L-functions attached to generalized Shimura lifts, which may be defined in a natural way since at almost all places the representations are unramified principal series.

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arXiv2015

On the Local Theory of Rankin-Selberg Convolutions for $\mathrm{SO_{2l}\times GL_{n}}$

The Rankin-Selberg method for studying Langlands' automorphic $L$-functions is to find integral representations, involving certain Fourier coefficients of cusp forms and Eisenstein series, for these functions. In this thesis we develop the local theory for generic representations of special orthogonal groups. We study the local integrals for $SO_{2l}\times GL_{n}$, where $SO_{2l}$ is the special even orthogonal group, either split or quasi-split, over a local non-Archimedean field. These integrals admit a functional equation, which is used to define a $γ$-factor. We show that, as expected, the $γ$-factor is identical with Shahidi's $γ$-factor. The analytic properties of the integrals are condensed into a notion of a greatest common divisor (g.c.d.). We establish certain bounds on the g.c.d. and relate it to the $L$-function defined by Shahidi in several cases, thereby providing another point of view on the $L$-function, linking it to the poles of the integrals. In particular, in the tempered case under a reasonable assumption the g.c.d. is equal to the $L$-function. Finally, this study includes the computation of the integrals with unramified data. This work may lead to new applications of the descent method, as well as aid in analyzing the poles of the global $L$-function.

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arXiv2015

The characterization of theta-distinguished representations of $GL_n$

Let $θ$ and $θ'$ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP], of a metaplectic double cover of $GL_n$. The tensor $θ\otimesθ'$ is a (very large) representation of $GL_n$. We characterize its irreducible generic quotients. In the square-integrable case, these are precisely the representations whose symmetric square L-function has a pole at s=0. Our proof of this case involves a new globalization result. In the general case these are the representations induced from distinguished data or pairs of representations and their contragradients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of $θ$. As a corollary, $θ$ is shown to admit a new "metaplectic Shalika model".

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arXiv2014

Theta distinguished representations, inflation and the symmetric square L-function

Let $Π_0$ be a representation of a group $H$. We say that a representation $τ$ is $(H,Π_0)$-distinguished, if it is a quotient of $Π_0$. It is natural to ask whether this notion "inflates" to larger groups, in the sense that a representation $\mathrm{I}(τ)$ induced from $τ$ and $H$ to a group $G$, is $(G,Π)$-distinguished. We study representations distinguished by theta representations: $H=GL_n$, $Π_0$ is a pair of the exceptional representations of Kazhdan and Patterson, $G=GSpin_{2n+1}$ and $Π$ is a pair of the small representations of Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation $τ$ is distinguished if and only if $\mathrm{I}(τ)$ is distinguished, and the multiplicity in each model is the same. If $τ$ is supercuspidal and distinguished, we prove that the Langlands quotient of $\mathrm{I}(τ)$ is distinguished. As a corollary, we characterize supercuspidal distinguished representations, in terms of the pole of the local symmetric square $L$-function at $s=0$.

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arXiv2014

The double cover of odd general spin groups, small representations and applications

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series, it is an automorphic representation and decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of SO(2n+1) of Bump, Friedberg and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin-Selberg integrals. We describe one application, to a calculation of a co-period integral.

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arXiv2014

Representations distinguished by pairs of exceptional representations and a conjecture of Savin

We study representations of GL(n) appearing as quotients of a tensor of exceptional representations, in the sense of Kazhdan and Patterson. Such representations are called distinguished. We characterize distinguished principal series representations in terms of their inducing data. In particular, we complete the proof of a conjecture of Savin, relating distinguished spherical representations to the image of the tautological lift from a suitable classical group.

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