arXiv subjects

Solomon Friedberg

Publications and source records attributed to Solomon Friedberg.

27 records · Page 2Linked to original sources

arXiv2015

Whittaker Coefficients of Metaplectic Eisenstein Series

We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However they are not Eulerian and the standard methods to compute them in the reductive case do not apply to covers. For "cominuscule" maximal parabolics, we give an explicit description of the coefficients as Dirichlet series whose arithmetic content is expressed in an exponential sum. The exponential sum is then shown to satisfy a twisted multiplicativity, reducing its determination to prime power contributions. These, in turn, are connected to Lusztig data for canonical bases on the dual group using a result of Kamnitzer. The exponential sum at prime powers is then evaluated for generic Lusztig data. To handle the remaining degenerate cases, the evaluation of the exponential sum appears best expressed in terms of string data for canonical bases, as shown in a detailed example in $GL_4$. Thus we demonstrate that the arithmetic part of metaplectic Whittaker coefficients is intimately connected to the relations between these two expressions for canonical bases.

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arXiv2014

Metaplectic Theta Functions and Global Integrals

We convolve a theta function on an $n$-fold cover of $GL_3$ with an automorphic form on an $n'$-fold cover of $GL_2$ for suitable $n,n'$. To do so, we induce the theta function to the $n$-fold cover of $GL_4$ and use a Shalika integral. We show that in particular when $n=n'=3$ this construction gives a new Eulerian integral for an automorphic form on the 3-fold cover of $GL_2$ (the first such integral was given by Bump and Hoffstein), and when $n=4$, $n'=2$, it gives a Dirichlet series with analytic continuation and functional equation that involves both the Fourier coefficients of an automorphic form of half-integral weight and quartic Gauss sums. The analysis of these cases is based on the uniqueness of the Whittaker model for the local exceptional representation. The constructions studied here may be put in the context of a larger family of global integrals which are constructed using automorphic representations on covering groups. We sketch this wider context and some related conjectures.

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arXiv2013

Eisenstein Series on Covers of Odd Orthogonal Groups

We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the $n$-fold cover of the split odd orthogonal group $SO_{2r+1}$. If the degree of the cover is odd, then Beineke, Brubaker and Frechette have conjectured that the $p$-power contributions to the Whittaker coefficients may be computed using the theory of crystal graphs of type C, by attaching to each path component a Gauss sum or a degenerate Gauss sum depending on the fine structure of the path. We establish their conjecture using a combination of automorphic and combinatorial-representation-theoretic methods. Surprisingly, we must make use of the type A theory, and the two different crystal graph descriptions of Brubaker, Bump and Friedberg available for type A based on different factorizations of the long word into simple reflections. We also establish a formula for the Whittaker coefficients in the even degree cover case, again based on crystal graphs of type C. As a further consequence, we establish a Lie-theoretic description of the coefficients for $n$ sufficiently large, thereby confirming a conjecture of Brubaker, Bump and Friedberg.

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arXiv2010

Metaplectic Ice

Spherical Whittaker functions on the metaplectic n-fold cover of GL(r+1) over a nonarchimedean local field containing n distinct n-th roots of unity may be expressed as the partition functions of statistical mechanical systems that are variants of the six-vertex model. If n=1 then in view of the Casselman-Shalika formula this fact is related to Tokuyama's deformation of the Weyl character formula. It is shown that various properties of these Whittaker functions may be expressed in terms of the commutativity of row transfer matrices for the system. Potentially these properties (which are already proved by other methods, but very nontrivial) are amenable to proof by the Yang-Baxter equation.

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arXiv2010

Schur Polynomials and the Yang-Baxter equation

We show that within the six-vertex model there is a parametrized Yang-Baxter equation with nonabelian parameter group GL(2)xGL(1) at the center of the disordered regime. As an application we rederive deformations of the Weyl character formule of Tokuyama and of Hamel and King.

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arXiv2006

On the p-parts of quadratic Weyl group multiple Dirichlet series

Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, which has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. Two constructions of such series are available, one based on summing products of n-th order Gauss sums, the second based on averaging a certain group action over the Weyl group. In this paper we study these constructions and the relationship between them, and give evidence that when n=2 and Phi=A_r they yield the same multiple Dirichlet series.

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arXiv1996

On local coefficients for non-generic representations of some classical groups

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier's theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Last, local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal.

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arXiv1995

The Fundamental Lemma for the Shalika Subgroup of GL$(4)$

The authors establish the fundamental lemma for a relative trace formula. This trace formula compares generic automorphic representations of GSp$(4)$ with automorphic representations of GL$(4)$ which are distinguished with respect to a character of the Shalika subgroup, the subgroup of matrices of $2\times 2$ block form $$ \pmatrix g&0\\0&g\endpmatrix \pmatrix I&0\\X&O\endpmatrix. $$ The fundamental lemma, giving the equality of two orbital integrals, amounts to a comparison of certain exponential sums arising from these two different groups. On the GSp$(4)$ side one has the Kloosterman sums for this group, and on the GL$(4)$ side certain, new, relative Kloosterman sums. To show that these are equal for each relevant Weyl group element, the authors compute the Mellin transforms of the sums and match them in all cases. The file available from the MSRI preprint server gives the first chapter of the manuscript. The entire manuscript, 147 pages long, is available as an MSRI preprint.

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arXiv1994

Explicit Formulas for the Waldspurger and Bessel Models

In this paper we will study certain models of irreducible admissible representations of the split special orthogonal group $SO(2n+1)$ over a nonarchimedean local field. If $n=1$, these models were considered by Waldspurger. If $n=2$, they were considered by Novodvorsky and Piatetski-Shapiro \cite{NP}, who called them {\it Bessel models}. They arise from a variety of Rankin-Selberg integrals nad the resultos of this paper will naturally have applications to the study of L-functions. They also arise in the study of the theta correspondence between $SO(2n+1)$ and the double cover of $Sp(2n)$, and they will therefore be of importance in generalizing the work of Waldspurger. As a global application we consider the Eisenstein series on $SO(2n+1)$ formed with a cuspidal automorphic representation $π$ on $GL(n)$, and we show that its Bessel period (6.2) is essentially a product of L-series. This generalizes work of Böcherer and Mizumoto.

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