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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk


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1435-5345
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On the p-parts of quadratic Weyl group multiple Dirichlet series

Gautam Chinta1 / Solomon Friedberg2 / Paul E. Gunnells3

1Department of Mathematics, The City College of CUNY, New York, NY 10031, USA. e-mail: chinta@sci.ccny.cuny.edu

2Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA. e-mail: friedber@bc.edu

3Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA. e-mail: gunnells@math.umass.edu

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2008, Issue 623, Pages 1–23, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2008.070, July 2008

Publication History

Received:
2006-12-25
Published Online:
2008-07-01

Abstract

Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s 1,…, sr, initially converging for ℜ(si) sufficiently large, which has meromorphic continuation to ℂr and satisfies functional equations under the transformations of ℂr corresponding to the Weyl group of Φ. Two constructions of such series are available, one [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series I, in: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, S. Friedberg, D. Bump, D. Goldfeld, and J. Hoffstein, eds., Proc. Symp. Pure Math. 75 (2006), 91–114.] [B. Brubaker, D. Bump, and S. Friedberg, Twisted Weyl group multiple Dirichlet series: the stable case, in: Eisenstein Series and Applications, Gan, Kudla, and Tschinkel, eds., Progr. Math. 258 (2008), 1–26.] [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] [B. Brubaker, D. Bump, and S. Friedberg, Weyl group multiple Dirichlet series II, The stable case, Invent. Math. 165 (2006), no. 2, 325–355.] based on summing products of n-th order Gauss sums, the second [G. Chinta and P. E. Gunnells, Weyl group multiple Dirichlet series constructed from quadratic characters, Invent. Math. 167 (2007), no. 2, 327–353.] based on averaging a certain group action over the Weyl group. In each case, the essential work occurs at a generic prime p; the local factors, satisfying local functional equations, are then pieced into a global object. In this paper we study these constructions and the relationship between them. First we extend the averaging construction to obtain twisted Weyl group multiple Dirichlet series, whose p-parts are given by evaluating certain rational functions in r variables. Then we develop properties of such a rational function, giving its precise denominator, showing that the nonzero coefficients of its numerator are indexed by points that are contained in a certain convex polytope, determining the coefficients corresponding to the vertices, and showing that in the untwisted case the rational function is uniquely determined from its polar behavior and the local functional equations. We also give evidence that in the case Φ = Ar, the p-part obtained here exactly matches the p-part of the twisted multiple Dirichlet series introduced in [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] when n = 2.

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[1]
Ben Brubaker, Daniel Bump, and Solomon Friedberg
Annals of Mathematics, 2011, Volume 173, Number 2, Page 1081

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