Summary
Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg, and Hoffstein [2]. Brubaker, Bump, and Friedberg [4] provided for when n is sufficiently large; the coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide with Whittaker coefficients of metaplectic Eisenstein series, but they are studied in these papers by a method that is independent of this fact. The assumption that n is large is called stability and allows a simple description of the Dirichlet series. “Twisted” Dirichet series were introduced in Brubaker, Bump, Friedberg, and Hoffstein [5] without the stability assumption, but only for root systems of type A{inr}. Their description is given differently, in terms of Gauss sums associated to Gelfand-Tsetlin patterns. In this paper, we reimpose the stability assumption and study the twisted multiple Dirichlet series for general Φ by introducing a description of the coefficients in terms of the root system similar to that given in the untwisted case in [4]. We prove the analytic continuation and functional equation of these series, and when Φ = A{inr} we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for the stable case.
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© 2008 Birkhäuser Boston
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Brubaker, B., Bump, D., Friedberg, S. (2008). Twisted Weyl Group Multiple Dirichlet Series: The Stable Case. In: Gan, W., Kudla, S., Tschinkel, Y. (eds) Eisenstein Series and Applications. Progress in Mathematics, vol 258. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4639-4_1
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DOI: https://doi.org/10.1007/978-0-8176-4639-4_1
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4496-3
Online ISBN: 978-0-8176-4639-4
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