Fourier expansions of KacMoody Eisenstein series and degenerate Whittaker vectors
Abstract
Motivated by string theory scattering amplitudes that are invariant under a discrete Uduality, we study Fourier coefficients of Eisenstein series on KacMoody 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 1/2BPS and 1/4BPS instantons, respectively. This suggests that there exist minimal and nexttominimal unipotent automorphic representations of the associated KacMoody 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 DOI:
 10.48550/arXiv.1312.3643
 arXiv:
 arXiv:1312.3643
 Bibcode:
 2013arXiv1312.3643F
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Number Theory;
 Mathematics  Representation Theory
 EPrint:
 62 pages. Journal version