We study the Weyl groups of hyperbolic Kac-Moody algebras of `over-extended type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed division algebras K=R,C,H,O, respectively. A crucial role is played by integral lattices of the division algebras and associated discrete matrix groups. Our findings can be summarized by saying that the even subgroups, W^+, of the Kac-Moody Weyl groups, W, are isomorphic to generalized modular groups over K for the simply laced algebras, and to certain finite extensions thereof for the non-simply laced algebras. This hints at an extended theory of modular forms and functions.
We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand-Kirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simply-laced groups, we give complete proofs only for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also use our formalism to calculate various local (real and p-adic) spherical vectors of minimal representations of the exceptional groups E_6, E_7, E_8 using global (adelic) degenerate Whittaker vectors, correctly reproducing existing results for such spherical vectors obtained by very different methods.
In this paper we investigate in detail the correspondence between E10 and Romans massive deformation of type IIA supergravity. We analyse the dynamics of a non-linear sigma model for a spinning particle on the coset space E10/K(E10) and show that it reproduces the dynamics of the bosonic as well as the fermionic sector of the massive IIA theory, within the standard truncation. The mass deformation parameter corresponds to a generator of E10 outside the realm of the generators entering the usual D=11 analysis, and is naturally included without any deformation of the coset model for E10/K(E10). Our analysis thus provides a dynamical unification of the massless and massiv
We compare the dynamics of maximal three-dimensional gauged supergravity in appropriate truncations with the equations of motion that follow from a one-dimensional E10/K(E10) coset model at the first few levels. The constant embedding tensor, which describes gauge deformations and also constitutes an M-theoretic degree of freedom beyond eleven-dimensional supergravity, arises naturally as an integration constant of the geodesic model. In a detailed analysis, we find complete agreement at the lowest levels. At higher levels there appear mismatches, as in previous studies. We discuss the origin of these mismatches.
The bosonic sector of various supergravity theories reduces to a homogeneous space G/H in three dimensions. The corresponding algebras g are simple for (half-)maximal supergravity, but can be semi-simple for other theories. We extend the existing literature on the Kac-Moody extensions of simple Lie algebras to the semi-simple case. Furthermore, we argue that for N=2 supergravity the simple algebras have to be augmented with an su(2) factor.
We continue the study of the one-dimensional E10 coset model (massless spinning particle motion on E10/K(E10) whose dynamics at low levels is known to coincide with the equations of motion of maximal supergravity theories in appropriate truncations. We show that the coset dynamics (truncated at levels less or equal to three) can be consistently restricted by requiring the vanishing of a set of constraints which are in one-to-one correspondence with the canonical constraints of supergravity. Hence, the resulting constrained sigma-model dynamics captures the full (constrained) supergravity dynamics in this truncation. Remarkably, the bosonic constraints are found to be expressible in a Sugawara-like (current x current) form in terms of the conserved E10 Noether current, and transform covariantly under an upper parabolic subgroup E10+ of E10. We discuss the possible implications of this result, and in particular exhibit a tantalising link with the usual affine Sugawara construction in the truncation of E10 to its affine subgroup E9.
We analyse four-dimensional gravity in the presence of general curvature squared corrections and show that Ehlers SL(2,R) symmetry, which appears in the reduction of standard gravity to three dimensions, is preserved by the correction terms. The mechanism allowing this is a correction of the SL(2,R) transformation laws which resolves problems with the different scaling behaviour of various terms occurring in the reduction.
