TY - JOUR

T1 - Coefficients of the n-fold theta function and Weyl group multiple Dirichlet series

AU - Brubaker, Benjamin

AU - Bump, Daniel

AU - Friedberg, Solomon

AU - Hoffstein, Jeffrey

PY - 2012

Y1 - 2012

N2 - We establish a link between certain Whittaker coefficients of the generalized metaplectic theta functions, first studied by Kazhdan and Patterson [Kazhdan and Patterson, Metaplectic forms, Inst. Hauteś Etudes Sci. Publ. Math., (59): 35-142, 1984], and the coefficients of stable Weyl group multiple Dirichlet series defined in [Brubaker, Bump, Friedberg,Weyl groupmultiple Dirichlet series. II. The stable case. Invent. Math., 165(2):325-355, 2006]. The generalized theta functions are the residues of Eisenstein series on a metaplectic n-fold cover of the general linear group. For n sufficiently large, we consider different Whittaker coefficients for such a theta function which lie in the orbit of Hecke operators at a given prime p. These are shown to be equal (up to an explicit constant) to the p-power supported coefficients of aWeyl group multiple Dirichlet series (MDS). These MDS coefficients are described in terms of the underlying root system; they have also recently been identified as the values of a p-adic Whittaker function attached to an unramified principal series representation on the metaplectic cover of the general linear group.

AB - We establish a link between certain Whittaker coefficients of the generalized metaplectic theta functions, first studied by Kazhdan and Patterson [Kazhdan and Patterson, Metaplectic forms, Inst. Hauteś Etudes Sci. Publ. Math., (59): 35-142, 1984], and the coefficients of stable Weyl group multiple Dirichlet series defined in [Brubaker, Bump, Friedberg,Weyl groupmultiple Dirichlet series. II. The stable case. Invent. Math., 165(2):325-355, 2006]. The generalized theta functions are the residues of Eisenstein series on a metaplectic n-fold cover of the general linear group. For n sufficiently large, we consider different Whittaker coefficients for such a theta function which lie in the orbit of Hecke operators at a given prime p. These are shown to be equal (up to an explicit constant) to the p-power supported coefficients of aWeyl group multiple Dirichlet series (MDS). These MDS coefficients are described in terms of the underlying root system; they have also recently been identified as the values of a p-adic Whittaker function attached to an unramified principal series representation on the metaplectic cover of the general linear group.

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U2 - 10.1007/978-1-4614-1219-9_4

DO - 10.1007/978-1-4614-1219-9_4

M3 - Article

AN - SCOPUS:84876214590

VL - 9

SP - 83

EP - 95

JO - Springer Proceedings in Mathematics

JF - Springer Proceedings in Mathematics

SN - 2190-5614

ER -