Arrangements of hyperplanes II: Szenes formula and Eisenstein series

1999/03/30 by Michel Brion, Brion, Michel, Michele Vergne +1
Mathematics · #11B68 #40B05 #52B30 #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:11B68 #msc:40B05 #msc:52B30

paper · pdf · doi:10.48550/arxiv.math/9903180

revised version (introduction rewritten, references added, minor changes made), 28 pages, LaTEX2e

arxiv created 1999/10/12 · arxiv updated 2009/11/30

Abstract

The aim of this article is to generalize in several variables some formulae for Eisenstein series in one variable. For example the formula 2ζ(2k) = (2π)^2k \fracB_2k(2k)! = Res_z=0(\frac1z^2k(1-e^z)) for the values of zeta functions at even integers in functions of Bernoulli numbers. A. Szenes proved in several variables a similar residue formula for the values of the zeta function introduced by Witten. We introduce some Eisenstein series by averaging over a lattice rational functions with poles in an arrangement of hyperplanes. We give another proof of Szenes residue formula by relating it to the constant term of these Eisenstein series.

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