A Crystal Definition for Symplectic Multiple Dirichlet Series

Beineke, Jennifer; Brubaker, Benjamin; Frechette, Sharon

doi:10.1007/978-0-8176-8334-4_2

Cited by 12 publications

(32 citation statements)

References 12 publications

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“…First, in Subsection 6.1 we shall rewrite the inductive formula of Theorem 1 in the p-power case as a sum over such top rows. Second, in Subsection 6.2 we shall recall the conjectural formula of Beineke, Brubaker and Frechette [3], which constructs certain coefficients H BZL directly from the crystal graph. In Section 7 we shall show that these coefficients in fact match the coefficients H of the Eisenstein series, as conjectured.…”

Section: Combinatorial Descriptions Of the Coefficients At Powers Of Pmentioning

confidence: 99%

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Eisenstein series on covers of odd orthogonal groups

Friedberg¹,

Zhang²

2015

ajm

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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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Section: Combinatorial Descriptions Of the Coefficients At Powers Of Pmentioning

confidence: 99%

“…For Eisenstein series on odd degree covers of a split orthogonal group of Cartan type B, a conjectural description of these terms using crystal graphs was given by Beineke, Brubaker and Frechette [3,4]. The quantum group that appears should be regarded as attached to the quantized Lie algebra of the L-group.…”

Section: Introductionmentioning

confidence: 99%

Eisenstein series on covers of odd orthogonal groups

Friedberg¹,

Zhang²

2015

ajm

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“…It is possible regard (4) as a multiple Dirichlet series, and indeed both (3) and (4) are treated together in Diaconu, Goldfeld and Hoffstein [32]. See [31] in this volume for a discussion of the sixth integral moment and its connection with the spectral theory of Eisenstein series on GL 3 . Returning to (3), there are two problems: to make a correct definition of the multiple Dirichlet series, and to determine its analytic properties.…”

Section: Andmentioning

confidence: 99%

“…See [31] in this volume for a discussion of the sixth integral moment and its connection with the spectral theory of Eisenstein series on GL 3 . Returning to (3), there are two problems: to make a correct definition of the multiple Dirichlet series, and to determine its analytic properties. If k = 1, 2 or 3, these can both be solved, and the multiple Dirichlet series has global analytic continuation.…”

Section: Andmentioning

confidence: 99%

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Introduction: Multiple Dirichlet Series

Bump

2012

Multiple Dirichlet Series, L-Functions and Automorphic Forms

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Crystal Constructions in Number Theory

Puskás

2019

Association for Women in Mathematics Series

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Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics.Date: October 16, 2018.

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A Crystal Definition for Symplectic Multiple Dirichlet Series

Cited by 12 publications

References 12 publications

Eisenstein series on covers of odd orthogonal groups

Eisenstein series on covers of odd orthogonal groups

Introduction: Multiple Dirichlet Series

Crystal Constructions in Number Theory

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