Date of Award
Open Access Dissertation
Doctor of Philosophy (PhD)
Paul E. Gunnells
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 . 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  to all _ and n. Additionally, we show that the p-parts of Chinta and Gunnells  agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg [4, 5] (in the cases when both constructions apply).
Friedlander, Holley Ann, "Twisted Weyl Group Multiple Dirichlet Series Over the Rational Function Field" (2013). Open Access Dissertations. 831.