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Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. (English) Zbl 1302.11032

This important paper establishes part of the Eisenstein conjecture. Specifically it is proved that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of \(\mathrm{GL}_{r+1}\) can be expressed in terms of Weyl group multiple Dirichlet series in \(r\) complex variables. To construct these Weyl group multiple Dirichlet series, the authors first attach quantities of number-theoretic nature to the vertices of a crystal graph. The “\(p\)-parts” of the coefficients of the Weyl group multiple Dirichlet series are then derived from the sum over those vertices.
In contrast to the non-metaplectic case, the coefficients of the Weyl group multiple Dirichlet series considered in the paper are not multiplicative. However, the authors prove that they satisfy a “twisted multiplicativity” property which suffices for determining the coefficients from their “\(p\)-parts”. The expression for the \(p\)-parts of these coefficients established in the paper can be considered as an extension of the formula of W. Casselman and J. Shalika [Compos. Math. 41, 207–231 (1980; Zbl 0472.22005)] and is thus of independent interest.
The authors show that the crystal graph description they give is equivalent to the Geldfand-Tsetlin description previously conjectured by the authors and J. Hoffstein [Ann. Math. (2) 166, No. 1, 293–316 (2007; Zbl 1154.11016)]. The \(p\)-parts of the coefficients were also described by G. Chinta and P. E. Gunnells [Invent. Math. 167, No. 2, 327–353 (2007; Zbl 1203.11062)] using averaging. Subsequent work by G. Chinta and O. Offen [Am. J. Math. 135, No. 2, 403–441 (2013; Zbl 1294.22012)] and by P. J. McNamara [Duke Math. J. 156, No. 1, 1–31 (2011; Zbl 1217.22015)] implies that the crystal graph description given in the paper under review agrees, in the case of type A, with the description given by Chinta and Gunnells [loc. cit.].

MSC:

11F68 Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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