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

Abstract

Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, 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 [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 R 4 and ∂ 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.

On supersymmetric E11 exceptional field theory

“…For the continuous E 11 , one finds immediately that only a two-derivative Lagrangian can possibly be invariant. Whether E 11 (Z) and its automorphic forms [100,101] can be used for higher derivative terms remains to be seen.…”

Fourier coefficients attached to small automorphic representations of SLn(A)

“…of [FKP14]. The power of this formula lies in that it expresses a degenerate Whittaker function evaluated on the Cartan torus of a group G(A) as a sum of generic Whittaker functions on a subgroup G ′ (A).…”

“…The Fourier coefficients of such automorphic forms therefore have a direct physical interpretation: the constant terms encode perturbative quantum corrections, while the non-constant terms correspond to non-perturbative, instanton, effects [FK12,FKP14,BV14,BV15a,BV15b,BCHP17b,BP17,BCHP17a]. For a recent book on automorphic representations and the connection with string theory, see [FGKP18].…”

“…where Π ′ is the set of simple roots of G ′ and w c is hence the summation variable and corresponds to a specific representative of the quotient Weyl group W W ′ described in [FKP14]. ρ denotes the Weyl vector, M denotes the intertwiner M (w, λ) = α>0 wα<0 ξ(⟨λ α⟩) ξ(⟨λ α⟩ + 1) as featured in the Langlands constant term formula, where ξ is the completed Riemann zeta function and ψ a denotes the "twisted character" both defined in appendix A.…”

“…For the special cases of SL 3 and SL 4 the Miller-Sahi results were generalized in [GKP16] (following related results in [FKP14]) to automorphic forms attached to a next-to-minimal automorphic representation π ntm . It was shown that any Fourier coefficient is completely determined by (1.1) and coefficients of the following form…”