Twisted Weyl Group Multiple Dirichlet Series Over the Rational Function Field

TWISTED WEYL GROUP MULTIPLE DIRICHLET SERIES OVER THE RATIONAL FUNCTION FIELD SEPTEMBER 2013 HOLLEY A. FRIEDLANDER, B.A., UNIVERSITY OF VERMONT M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor 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 Φ be an irreducible, reduced root system, and let n be an integer greater than 1. Fix a prime power q ∈ 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 Φ 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