Whittaker Functions

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Eisenstein Series ◽  
Local Field ◽  
Whittaker Function ◽  
Global Field ◽  
Whittaker Functions ◽  
Schur Polynomials ◽  
Multiple Dirichlet Series ◽  
Adele Group

This chapter shows that Weyl group multiple Dirichlet series are expected to be Whittaker coefficients of metaplectic Eisenstein series. The fact that Whittaker coefficients of Eisenstein series reduce to the crystal description that was given in Chapter 2 is proved for Type A. On the adele group, the corresponding local computation reduces to the evaluation of a type of λ‎-adic integral. These were considered by McNamara, who reduced the integrals to sums over crystals by a very interesting method. A full treatment of this topic is outside the scope of this work, but it is introduced in this chapter by considering the case where n = 1. In this chapter, F is used to denote a nonarchimedean local field and F to denote a global field. The values of the Whittaker function are Schur polynomials multiplied by the normalization constant.

scholarly journals Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar

2007 ◽  
Vol 166 (1) ◽  
pp. 293-316 ◽  
Author(s):  
Benjamin Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg ◽  
Jeffrey Hoffstein
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Eisenstein Series ◽  
Multiple Dirichlet Series

scholarly journals Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

2011 ◽  
Vol 173 (2) ◽  
pp. 1081-1120 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Eisenstein Series ◽  
Crystal Bases ◽  
Multiple Dirichlet Series

Statement B and the Yang-Baxter Equation

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Statistical Mechanics ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Transfer Matrix ◽  
Baxter Equation ◽  
Partition Functions ◽  
Alternate Proof ◽  
Crystal Basis ◽  
Exactly Solved Models ◽  
Multiple Dirichlet Series

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

Tokuyama’s Theorem

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Eisenstein Series ◽  
Whittaker Function ◽  
Character Formula ◽  
Schur Polynomials ◽  
Normalization Constant ◽  
Small Letter ◽  
Weyl Character Formula

This chapter introduces the Tokuyama's Theorem, first by writing the Weyl character formula and restating Schur polynomials, the values of the Whittaker function multiplied by the normalization constant. The λ‎-parts of Whittaker coefficients of Eisenstein series can be profitably regarded as a deformation of the numerator in the Weyl character formula. This leads to deformations of the Weyl character formula. Tokuyama gave such a deformation. It is an expression of ssubscript Greek small letter lamda(z) as a ratio of a numerator to a denominator. The denominator is a deformation of the Weyl denominator, and the numerator is a sum over Gelfand-Tsetlin patterns with top row λ‎ + ρ‎.

Type A Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Algebraic Number ◽  
Type A ◽  
Algebraic Number Field ◽  
Basis Vector ◽  
Roots Of Unity ◽  
Finite Set ◽  
Multiple Dirichlet Series

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Riemann Zeta Function ◽  
Functional Equations ◽  
Lattice Points ◽  
Baxter Equation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Multiple Dirichlet Series

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's 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.

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

2008 ◽  
Vol 2008 (623) ◽  
pp. 1-23 ◽  
Author(s):  
Gautam Chinta ◽  
Solomon Friedberg ◽  
Paul E. Gunnells
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Multiple Dirichlet Series
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