Communications in Number Theory and Physics

Volume 8 (2014)

Number 1

Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors

Pages: 41 – 100



Philipp Fleig (Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Potsdam, Germany; and Freie Universität Berlin, Institut für Theoretische Physik, Berlin, Germany)

Axel Kleinschmidt (Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Potsdam, Germany; and International Solvay Institutes, Brussels, Belgium)

Daniel Persson (Fundamental Physics, Chalmers University of Technology, Gothenburg, Sweden)


Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac-Moody groups. In particular, we analyse the Eisenstein series on $E_{9}(\mathbb{R})$, $E_{10}(\mathbb{R})$ and $E_{11}(\mathbb{R})$ corresponding to certain degenerate principal series at the values $s = 3 / 2$ and $s = 5 / 2$ that were studied in [1]. We show that these Eisenstein series have very simple Fourier coefficients as expected for their role as supersymmetric contributions to the higher derivative couplings $\mathcal{R}^4$ and $\partial^4 \mathcal{R}^4$ coming from 1/2-BPS and 1/4-BPS instantons, respectively. This suggests that there exist minimal and next-to-minimal unipotent automorphic representations of the associated Kac-Moody groups to which these special Eisenstein series are attached. We also provide complete explicit expressions for degenerate Whittaker vectors of minimal Eisenstein series on $E_{6}(\mathbb{R})$, $E_{7}(\mathbb{R})$ and $E_{8}(\mathbb{R})$ that have not appeared in the literature before.

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