Constructing Weyl Group Multiple Dirichlet Series

This paper describes a technique to construct Weyl group multiple Dirichlet series. Such series were first introduced in [BBC06], which also described a heuristic means to define, analytically continue, and prove functional equations for a family of Dirichlet series in several complex variables. Several subsequent papers have dealt with the problem of making the heuristic definitions precise and completing the proofs of analytic continuation and functional equations of these Weyl group multiple Dirichlet series along the lines suggested in [BBC06]. Before listing some of the partial results obtained in these papers, we say a bit more about the type of multiple Dirichlet series studied. Let F be an algebraic number field containing the 2nth roots of unity. Fix a finite set of places S containing all the Archimedean places and those that are ramified over Q. Take S large enough that OS , the ring of S-integers of F , has class number one. Let Φ be a reduced root system of rank r. Let m = (m1, . . . ,mr) be an rtuple of integers in OS and s = (s1, . . . , sn) an r-tuple of complex variables. In (5.6) we define a certain finite-dimensional vector space M(Ω,Φ) of complex-valued functions on (F× S ) , and we choose Ψ ∈ M(Ω,Φ). In Section 4 we define coefficients H(c;m), where c and m are r-tuples of nonzero integers in OS . To this data we associate a multiple Dirichlet series in r complex variables