Weyl group multiple Dirichlet series. Type A combinatorial theory
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, these Dirichlet series are expected to have analytic continuation and to satisfy functional equations. By contrast, these Weyl group multiple Dirichlet series differ from usual L-functions in that they are usually functions of several complex variables, and in that their coefficients are multiplicative only up to roots of unity.
The group of symmetries of functional equations the series are arbitrary finite Weyl groups. This book is centered in the case of Weyl groups with Dynkin diagrams of type A. The coefficients of the Weyl group multiple Dirichlet 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 once it is proven that they are 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. This is accomplished through a series of surprising combinatorial reductions. The main bulk of the book (chapters 6 to 17, out of the 20 chapters in total) consists on the proof of this result.
The book also includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation. It is a technical piece of work addressed to experts in the area.