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Weyl group multiple Dirichlet series of type A 2

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Number Theory, Analysis and Geometry

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Abstract

A Weyl group multiple Dirichlet seriesis a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank rof the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group Wof Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n thorder Gauss sums attached to the root system of type A 2. The basic technique is that of [11, 10]; namely, we construct a rational function in rvariables invariant under a certain action of W, and use this to build a “local factor” of the global series.

In memory of Serge Lang

Mathematics Subject Classification (2010): Primary 11F68; Secondary 11F30

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Notes

  1. 1.

    We remark that our normalization for Gauss sums follows [3,  6] and not [11,  10]. See [11, Remark 3.12] for a discussion of this.

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Acknowledgements

GC wishes to thank the NSF for support of this research through the FRG grant DMS-0652605. GC also gratefully acknowledges the support of the Alexander von Humboldt Foundation.

PG wishes to thank the NSF for support through Grant DMS-0801214.

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Authors and Affiliations

  1. Department of Mathematics, The City College of CUNY, New York, NY, 10031, USA

    Gautam Chinta

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Paul E. Gunnells

Authors
  1. Gautam Chinta
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  2. Paul E. Gunnells
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Correspondence to Gautam Chinta .

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Editors and Affiliations

  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

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Chinta, G., Gunnells, P.E. (2012). Weyl group multiple Dirichlet series of type A 2 . In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_6

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