Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory

Preface An L-function, as the term is generally understood, is a Dirichlet series in one complex variable s with an Euler product that has (at least conjecturally) analytic continuation to all complex s and a functional equation under a single reflection s → 1 − s. The coefficients are in particular multiplicative. By contrast Weyl group multiple Dirichlet series are a new class of Dirichlet series with arithmetic content that differ from L-functions in two ways. First, although the coefficients of the series are not multiplicative in the usual sense, they are twisted multiplicative, the multiplicativity being modified by some n-th power residue symbols – see (1.3) below. Second, they are Dirichlet series in several complex variables s 1 , · · · , s r. They have (at least conjecturally) meromorphic continuation to all C r and groups of functional equations that are finite reflection groups. The data needed to define such a series in r complex variables are a root system Φ of rank r with Weyl group W , a fixed integer n > 1, and a global ground field F containing the n-th roots of unity; in some of the literature (including this work) the ground field F is assumed to contain the 2n-th roots of unity. Twisted multiplica-tivity implies that it is sufficient to describe the prime-power coefficients of such a series. In this work we consider the case that Φ is of Cartan type A r. In this case a class of multiple Dirichlet series, convergent for (s i) sufficiently large, was described in [10], where the analytic continuation and functional equations were conjectured. Their definition is given in detail in Chapter 1 below. The prime-power coefficients are sums of products of n-th order Gauss sums, with the individual terms indexed by Gelfand-Tsetlin patterns. It is not clear from this definition that these series have analytic continuation and functional equations. However it was shown in [9] that this global property would be a consequence of a conjectured purely local property of a combinatorial and number-theoretic nature. Specifically, two distinct versions of the Gelfand-Tsetlin definition were given. It is not apparent that they are equal. Either of these definitions is purely local in that it specifies the p-part of the multiple Dirichlet series, and this then determines the ii iii global Dirichlet series by twisted multiplicativity. It was proved in [9] that if these two …