Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

Abstract

We show that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of GLr+1 can be described as multiple Dirichlet series in r complex variables, whose coefficients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on "string data" previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reflections. The coefficients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

Schubert Eisenstein Series

“…The variety Z w is always nonsingular, and may be built up by successive fiberings by P 1 , which corresponds to the procedure in representation theory of reducing a computation on G to a series of SL 2 computations. And this is what was done (for the full Eisenstein series, that is, for the case where w = w 0 ) in Brubaker, Bump and Friedberg [3].…”

“…See the survey article Bump [5] for discussion of this this phenomenon and its history. An analysis of the proof of one particular case, in Brubaker, Bump and Friedberg [3] shows the mechanism behind this phenomenon makes use of Bott-Samelson varieties. In this connection, we call attention to one particular point: that such a representation of the Whittaker coefficient of an Eisenstein series as a sum over a crystal requires a choice of a reduced word, by which we mean a decomposition of the long Weyl group element w 0 into a product of simple reflections of shortest possible length.…”

“…(Proposition 13.) This method of representing the Eisenstein series E(g, ν) = E w 0 (g, ν), with w 0 the long Weyl group element, is implicit in the method used by Brubaker, Bump and Friedberg [3] in order to prove that the Whittaker function of Eisenstein series on the metaplectic cover of GL r+1 (F ) had a representation as a sum over a crystal basis of a representation of GL r+1 . The proof depends on a parametrization, described in Section 5 of the paper, of an element of P \G, where P is a maximal parabolic subgroup, by choosing the representative factored over such a product of SL 2 .…”

Coefficients of the n-Fold Theta Function and Weyl Group Multiple Dirichlet Series

“…Though we have described the series (1) in terms of global objects (Eisenstein series), let us also mention that the p-power supported terms are known to match the p-adic Whittaker function attached to the spherical vector for the associated principal series representation used to construct the Eisenstein series. This follows from combining the work of McNamara [17] with that of Brubaker, Bump and Friedberg [2,6], or by combining [2,17] and an unfolding argument of Friedberg and McNamara [11]. The precise definition of the coefficients H in the "stable" case will be reviewed in Section 2.…”

“…More recently, Brubaker, Bump and Friedberg [6] have given an explicit description of the Whittaker coefficients of Borel Eisenstein series on the n-fold metaplectic cover of GL r+1 . In particular, they showed that the first nondegenerate Whittaker coefficient is a Dirichlet series in r complex variables (a "multiple Dirichlet series") that is roughly of the form…”