Weyl Group Multiple Dirichlet Series and Gelfand-Tsetlin Patterns

An L-function, as the term is generally understood, is a Dirichlet series in one complex variable s with analyic continuation to all complex s, a functional equation and an Euler product. The coefficients are thus multiplicative. Weyl group multiple Dirichlet series are a new class of Dirichlet series that differ from L-functions in two ways. First, they are Dirichlet series in several complex variables s1, · · · , sr with meromorphic continuation to all C, whose groups of functional equations are finite reflection groups; second, although the coefficients are almost multiplicative, the multiplicativity is twisted by n-th power residue symbols, and the series are thus not Euler products. Conjecturally, these series can be identified with Whittaker coefficients of Eisenstein series on metaplectic groups, but current methods of study do not make use of this connection, which is less directly useful than the connection between L-functions and (nonmetaplectic) Eisenstein series in the Langlands-Shahidi method. One reason is that the metaplectic Eisenstein series in question usually do not have unique Whittaker models. Weyl group multiple Dirichlet series were introduced formally in Brubaker, Bump, Chinta, Friedberg and Hoffstein [4] but particular instances and related multiple Dirichlet series can be found in earlier papers such as Siegel [27], Goldfeld and Hoffstein [18], Kubota [21], Bump, Friedberg and Hoffstein [9], Chinta [11] and Peter [23], [24]. The essential data in describing such a series in r complex variables are a root system Φ of rank r with Weyl group W and a global ground field F containing the n-th roots of unity; in some of the literature (including this paper) the ground field F is assumed to contain the 2n-th roots of unity. The coefficients involve n-th order Gauss sums. The series has meromorphic continuation to all of C and a group of functional equations isomorphic to W .