Weyl group multiple Dirichlet series II: The stable case

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

“…In [3], Brubaker, Bump, and Friedberg defined a Weyl group multiple Dirichlet series for any reduced root system and for n sufficiently large (depending on the root system). The requirement that n be sufficiently large is called stability, as the coefficients of the Dirichlet series are uniformly described Lie-theoretically for all such n. In this paper we will be concerned with root systems of type A r and in this case, the stability condition is satisfied if n ≥ r.…”

“…However we will want to consider H independently of Ψ, so for each prime p of o S we fix a generator p of p, and only consider c i which are products of powers of these fixed p's. The function Ψ is chosen from a finite-dimensional vector space that is wellunderstood and defined in [3] or [2], and we will not discuss it further here. The function H contains the key arithmetic information.…”

“…The general expression for the coefficients H is best given in the language of crystal graphs, but this full description will not be needed here. Indeed, we will restrict our attention to cases where the degree of the cover n is at least r + 1 (which is "stable" in the vocabulary of [3]), in which case the description of the coefficients simplifies considerably. In this situation, the coefficients supported at powers of a given prime p are in one-to-one correspondence with the Weyl group S r+1 of GL r+1 , and have a description in terms of the underlying root system [3].…”

“…Indeed, we will restrict our attention to cases where the degree of the cover n is at least r + 1 (which is "stable" in the vocabulary of [3]), in which case the description of the coefficients simplifies considerably. In this situation, the coefficients supported at powers of a given prime p are in one-to-one correspondence with the Weyl group S r+1 of GL r+1 , and have a description in terms of the underlying root system [3]. 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.…”

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AbstractWe establish a link between certain Whittaker coefficients of the generalized metaplectic theta functions, first studied by Kazhdan and Patterson [14], and the coefficients of the stable Weyl group multiple Dirichlet series defined in [3]. The generalized theta functions are the residues of Eisenstein series on a metaplectic n-fold cover of the general linear group.

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