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Journal Article

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


Fleig,  Philipp
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, Golm, DE;


Kleinschmidt,  Axel
Quantum Gravity and Unified Theories, AEI Golm, MPI for Gravitational Physics, Max Planck Society;

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Fleig, P., Kleinschmidt, A., & Persson, D. (2014). Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors. Communications in Number Theory and Physics, 8(1), 41-100. doi:10.4310/CNTP.2014.v8.n1.a2.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0014-D1B4-8
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.