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

TL;DR

This paper investigates Fourier coefficients of Eisenstein series on Kac-Moody groups related to string theory, revealing simple structures linked to supersymmetric instanton contributions and providing explicit formulas for Whittaker vectors.

Contribution

It introduces explicit formulas for degenerate Whittaker vectors on certain Kac-Moody groups and analyzes Fourier coefficients of Eisenstein series relevant to string theory amplitudes.

Findings

Eisenstein series on E9, E10, E11 have simple Fourier coefficients.

These coefficients relate to supersymmetric higher derivative couplings.

Explicit expressions for Whittaker vectors on E6, E7, E8 are provided.

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 $E_9(R)$, $E_{10}(R)$ and $E_{11}(R)$ corresponding to certain degenerate principal series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show that these Eisenstein series have very simple Fourier coefficients as expected for their role as supersymmetric contributions to the higher derivative couplings $R^4$ and $\partial^{4} R^4$ 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 $E_6(R)$, $E_7(R)$ and $E_8(R)$ that have not appeared in the literature before.