Whittaker Coefficients of Metaplectic Eisenstein Series

Benjamin Brubaker, Solomon Friedberg

Research output: Contribution to journalArticle

9 Scopus citations

Abstract

We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However they are not Eulerian and the standard methods to compute them in the reductive case do not apply to covers. For “cominuscule” maximal parabolics, we give an explicit description of the coefficients as Dirichlet series whose arithmetic content is expressed in an exponential sum. The exponential sum is then shown to satisfy a twisted multiplicativity, reducing its determination to prime power contributions. These, in turn, are connected to Lusztig data for canonical bases on the dual group using a result of Kamnitzer. The exponential sum at prime powers is shown to simplify for generic Lusztig data. At the remaining degenerate cases, the exponential sum seems best expressed in terms of Gauss sums depending on string data for canonical bases, as shown in a detailed example in GL4. Thus we demonstrate that the arithmetic part of metaplectic Whittaker coefficients is intimately connected to the relations between these two expressions for canonical bases.

Original languageEnglish (US)
Pages (from-to)1180-1239
Number of pages60
JournalGeometric and Functional Analysis
Volume25
Issue number4
DOIs
StatePublished - Jul 21 2015

Keywords

  • Canonical bases
  • Eisenstein series
  • Lusztig data
  • Metaplectic group
  • Whittaker coefficient

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