Weyl Group Multiple Dirichlet Series

Ben Brubaker; Daniel Bump; Solomon Friedberg

Ben Brubaker, Daniel Bump and Solomon Friedberg

Weyl Group Multiple Dirichlet Series

Type A Combinatorial Theory (AM-175)

In: Annals of Mathematics Studies, 175

Princeton University Press | 2011

https://doi.org/10.1515/9781400838998

Overview
Contents

About this book

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics.

These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished.

The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.

Author information

Ben Brubaker is assistant professor of mathematics at Massachusetts Institute of Technology. Daniel Bump is professor of mathematics at Stanford University. Solomon Friedberg is professor of mathematics at Boston College.

Frontmatter Free access PDF PDF	i
Contents Free access PDF PDF	v
Preface Access restricted Content is available PDF PDF	vii
Chapter One. Type A Weyl Group Multiple Dirichlet Series Access restricted Content is available PDF PDF	1
Chapter Two. Crystals and Gelfand-Tsetlin Patterns Access restricted Content is available PDF PDF	10
Chapter Three. Duality Access restricted Content is available PDF PDF	22
Chapter Four. Whittaker Functions Access restricted Content is available PDF PDF	26
Chapter Five. Tokuyama’s Theorem Access restricted Content is available PDF PDF	31
Chapter Six. Outline of the Proof Access restricted Content is available PDF PDF	36
Chapter Seven. Statement B Implies Statement A Access restricted Content is available PDF PDF	51
Chapter Eight. Cartoons Access restricted Content is available PDF PDF	54
Chapter Nine. Snakes Access restricted Content is available PDF PDF	58
Chapter Ten. Noncritical Resonances Access restricted Content is available PDF PDF	64
Chapter Eleven. Types Access restricted Content is available PDF PDF	67
Chapter Twelve. Knowability Access restricted Content is available PDF PDF	74
Chapter Thirteen. The Reduction to Statement D Access restricted Content is available PDF PDF	77
Chapter Fourteen. Statement E Implies Statement D Access restricted Content is available PDF PDF	87
Chapter Fifteen. Evaluation of ΛΓ and ΛΔ, and Statement G Access restricted Content is available PDF PDF	89
Chapter Sixteen. Concurrence Access restricted Content is available PDF PDF	96
Chapter Seventeen. Conclusion of the Proof Access restricted Content is available PDF PDF	104
Chapter Eighteen. Statement B and Crystal Graphs Access restricted Content is available PDF PDF	108
Chapter Nineteen. Statement B and the Yang-Baxter Equation Access restricted Content is available PDF PDF	115
Chapter Twenty. Crystals and p-adic Integration Access restricted Content is available PDF PDF	132
Bibliography Access restricted Content is available PDF PDF	143
Notation Access restricted Content is available PDF PDF	149
Index Access restricted Content is available PDF PDF	155

(Deutsch)

Weyl Group Multiple Dirichlet Series

Type A Combinatorial Theory (AM-175)

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