Integer Forms of Kac–moody Groups and Eisenstein Series in Low Dimensional Supergravity Theories

Abstract. Kac–Moody groups G over R have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms G(Z) are conjecturally U– duality groups. Mathematical descriptions of G(Z), due to Tits, are functorial and not amenable to computation or applications. We construct Kac–Moody groups over R and Z using an analog of Chevalley’s constructions in finite dimensions and Garland’s constructions in the a ne case. We extend a construction of Eisenstein series on finite dimensional simple algebraic groups using representation theory, which appeared in the context of string theory, to general Kac– Moody groups. This coincides with a generalization of Garland’s Eisenstein series on a ne Kac–Moody groups to general Kac–Moody groups and includes Eisenstein series on E10 and E11. For finite dimensional groups, Eisenstein series encode the quantum corrections in string theory and supergravity theories. Their Kac–Moody analogs will likely also play an important part in string theory, though their roles are not yet understood.