Coefficients of the n-Fold Theta Function and Weyl Group Multiple Dirichlet Series
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Auteurs : Benjamin Brubaker [États-Unis] ; Daniel Bump [États-Unis] ; Solomon Friedberg [États-Unis] ; Jeffrey Hoffstein [États-Unis]Source :
- Springer Proceedings in Mathematics [ 2190-5614 ] ; 2012.
Abstract
Abstract: 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. Hautes Études Sci. Publ. Math., (59): 35–142, 1984], and the coefficients of stable Weyl group multiple Dirichlet series defined in [Brubaker, Bump, Friedberg, Weyl group multiple 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 a Weyl 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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DOI: 10.1007/978-1-4614-1219-9_4
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Hautes Études Sci. Publ. Math., (59): 35–142, 1984], and the coefficients of stable Weyl group multiple Dirichlet series defined in [Brubaker, Bump, Friedberg, Weyl group multiple 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 a Weyl 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.</div></front></TEI><affiliations><list><country><li>États-Unis</li></country><region><li>Californie</li><li>Massachusetts</li><li>Rhode Island</li></region><settlement><li>Providence (Rhode Island)</li></settlement><orgName><li>Université Brown</li></orgName></list><tree><country name="États-Unis"><region name="Massachusetts"><name sortKey="Brubaker, Benjamin" sort="Brubaker, Benjamin" uniqKey="Brubaker B" first="Benjamin" last="Brubaker">Benjamin Brubaker</name></region><name sortKey="Brubaker, Benjamin" sort="Brubaker, Benjamin" uniqKey="Brubaker B" first="Benjamin" last="Brubaker">Benjamin Brubaker</name><name sortKey="Bump, Daniel" sort="Bump, Daniel" uniqKey="Bump D" first="Daniel" last="Bump">Daniel Bump</name><name sortKey="Bump, Daniel" sort="Bump, Daniel" uniqKey="Bump D" first="Daniel" last="Bump">Daniel Bump</name><name sortKey="Friedberg, Solomon" sort="Friedberg, Solomon" uniqKey="Friedberg S" first="Solomon" last="Friedberg">Solomon Friedberg</name><name sortKey="Friedberg, Solomon" sort="Friedberg, Solomon" uniqKey="Friedberg S" first="Solomon" last="Friedberg">Solomon Friedberg</name><name sortKey="Hoffstein, Jeffrey" sort="Hoffstein, Jeffrey" uniqKey="Hoffstein J" first="Jeffrey" last="Hoffstein">Jeffrey Hoffstein</name><name sortKey="Hoffstein, Jeffrey" sort="Hoffstein, Jeffrey" uniqKey="Hoffstein J" first="Jeffrey" last="Hoffstein">Jeffrey Hoffstein</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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