Twisted Weyl Group Multiple Dirichlet Series Over The Rational Function Field
Download Twisted Weyl Group Multiple Dirichlet Series Over The Rational Function Field PDF/ePub or read online books in Mobi eBooks. Click Download or Read Online button to get Twisted Weyl Group Multiple Dirichlet Series Over The Rational Function Field book now. This website allows unlimited access to, at the time of writing, more than 1.5 million titles, including hundreds of thousands of titles in various foreign languages.
Twisted Weyl Group Multiple Dirichlet Series Over the Rational Function Field
Let K be a global field. For each prime p of K, the p-part of a multiple Dirichlet series defined over K is a generating function in several variables for the p-power coefficients. Let _ be an irreducible, reduced root system, and let n be an integer greater than 1. Fix a prime power q 2 Z congruent to 1 modulo 2n, and let Fq(T) be the field of rational functions in T over the finite field Fq of order q. In this thesis, we examine the relationship between Weyl group multiple Dirichlet series over K = Fq(T) and their p-parts, which we define using the Chinta-Gunnells method [10]. Our main result shows that Weyl group multiple Dirichlet series of type _ over Fq(T) may be written as the finite sum of their p-parts (after a certain variable change), with "multiplicities" that are character sums. This result gives an analogy between twisted Weyl group multiple Dirichlet series over the rational function field and characters of representations of semi-simple complex Lie algebras associated to _. Because the p-parts and global series are closely related, the result above follows from a series of local results concerning the p-parts. In particular, we give an explicit recurrence relation on the coefficients of the p-parts, which allows us to extend the results of Chinta, Friedberg, and Gunnells [9] to all _ and n. Additionally, we show that the p-parts of Chinta and Gunnells [10] agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg [4, 5] (in the cases when both constructions apply).
Weyl Group Multiple Dirichlet Series
Author: Ben Brubaker
language: en
Publisher: Princeton University Press
Release Date: 2011-07-05
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.
Multiple Dirichlet Series, L-functions and Automorphic Forms
Multiple Dirichlet Series, L-functions and Automorphic Forms gives the latest advances in the rapidly developing subject of Multiple Dirichlet Series, an area with origins in the theory of automorphic forms that exhibits surprising and deep connections to crystal graphs and mathematical physics. As such, it represents a new way in which areas including number theory, combinatorics, statistical mechanics, and quantum groups are seen to fit together. The volume also includes papers on automorphic forms and L-functions and related number-theoretic topics. This volume will be a valuable resource for graduate students and researchers in number theory, combinatorics, representation theory, mathematical physics, and special functions. Contributors: J. Beineke, B. Brubaker, D. Bump, G. Chinta, G. Cornelissen, C.A. Diaconu, S. Frechette, S. Friedberg, P. Garrett, D. Goldfeld, P.E. Gunnells, B. Heim, J. Hundley, D. Ivanov, Y. Komori, A.V. Kontorovich, O. Lorscheid, K. Matsumoto, P.J. McNamara, S.J. Patterson, M. Suzuki, H. Tsumura.