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# 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

DOI: http://dx.doi.org/10.4310/CNTP.2014.v8.n1.a2

#### Authors

#### Abstract

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.