Hyperbolic Kac-Moody groups and algebras

Kac-Moody algebras are the most natural generalization to infinite dimensions of finite dimensional simple Lie algebras. The the commutator subalgebras of affine Kac-Moody algebras are central extensions of loop algebras and affine Kac-Moody groups are central extension of classical Lie groups. However for hyperbolic and Lorentzian Kac-Moody groups and algebras many fundamental questions about their structure remain unanswered.

The class of hyperbolic and Lorentzian Kac-Moody algebras and their associated groups has not yet been extensively studied by mathematicians: not for lack of interest or lack of mathematical richness, but rather because of the number of difficult and persistent fundamental open problems regarding their structure. Indeed these infinite dimensional Lie groups and algebras promise to give rise to intricate modular forms and their internal structures carry deep and beautiful algebraic and geometric properties.

Hyperbolic and Lorentzian Kac-Moody groups and algebras have recently been discovered to occur as symmetries in high energy theoretical physics, though their role is not well understood. There is evidence to suggest that these groups and algebras appear as symmetries of dimensionally reduced supergravity. Of particular importance are the symmetries of discrete forms G(Z) of Kac-Moody groups G and automorphic forms associated to these groups.

Due to the nature and status of the subject, in recent collaborations we have approached this subject on all levels: abstract, concrete, computational. Since many of the interesting mathematical questions are physically motivated, in collaborations with physicists we consider these questions together with their conjectural physical symmetries.

In order to work with discrete symmetries, we require a convenient description of discrete forms G(Z) of Kac-Moody groups G. Mathematical descriptions of G(Z) for Kac-Moody groups G due to Tits, are functorial and not amenable to computation or applications. We have constructed Kac-Moody groups over R and Z using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We have also determined generators, relations and structure constants for computation in the hyperbolic Kac-Moody groups of interest.

To provide further techniques for applications, we have determined the structure of orbits of Weyl groups of Kac-Moody algebras on the the real roots of their root systems. We revisited and corrected the classification of hyperbolic Dynkin diagrams and gave a detailed description of the simple root lengths and Weyl group orbits of hyperbolic Kac-Moody root systems.

To begin our study of automorphic forms on Kac-Moody groups, we have constructed Eisenstein series on arithmetic quotients of rank 2 Kac-Moody groups over finite fields and we proved meromprphic continuation. We extended a construction of Eisenstein series using representation theory to general Kac-Moody groups over R. We have also constructed congruence subgroups of lattices in rank 2 Kac-Moody groups over finite fields.

To investigate the internal structure of Kac-Moody groups, we showed that Kac-Moody groups over finite fields contain nonuniform lattice subgroups, and we constructed uniform lattices in low rank. We showed that rank 2 hyperbolic Kac-Moody algebras give rise to an infinite sequence of Fibonacci type integer sequences that have relationships with real quadratic fields and the cohomology of flag varieties of rank 2 Kac-Moody groups over C. We proved that complete Kac-Moody groups over finite fields satisfy a simplicity theorem of Tits. We showed that certain Kac-Moody groups over finite fields satisfy the Baum-Connes conjecture in non-commutative geometry that relates the analytic and topological properties of a group.