Weyl group multiple Dirichlet / Ben Brubaker, Daniel Bump, and Solomon Friedberg.

Format:

Book

Author/Creator:

Brubaker, Ben, 1976-

Contributor:

Bump, Daniel, 1952-

Friedberg, Solomon, 1958-

Series:

Annals of mathematics studies ; no. 175.

Annals of mathematics studies ; no. 175

Language:

English

Subjects (All):

Dirichlet series.

Weyl groups.

Physical Description:

1 online resource (173 p.)

Edition:

Course Book

Place of Publication:

Princeton, N.J. : Princeton University Press, c2011.

Language Note:

English

Summary:

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.

Contents:

Front matter

Contents

Preface

Chapter One. Type A Weyl Group Multiple Dirichlet Series

Chapter Two. Crystals and Gelfand-Tsetlin Patterns

Chapter Three. Duality

Chapter Four. Whittaker Functions

Chapter Five. Tokuyama's Theorem

Chapter Six. Outline of the Proof

Chapter Seven. Statement B Implies Statement A

Chapter Eight. Cartoons

Chapter Nine. Snakes

Chapter Ten. Noncritical Resonances

Chapter Eleven. Types

Chapter Twelve. Knowability

Chapter Thirteen. The Reduction to Statement D

Chapter Fourteen. Statement E Implies Statement D

Chapter Fifteen. Evaluation of ΛΓ and ΛΔ, and Statement G

Chapter Sixteen. Concurrence

Chapter Seventeen. Conclusion of the Proof

Chapter Eighteen. Statement B and Crystal Graphs

Chapter Nineteen. Statement B and the Yang-Baxter Equation

Chapter Twenty. Crystals and p-adic Integration

Bibliography

Notation

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786613013385

9781283013383

128301338X

9781400838998

1400838991

OCLC:

713258718