AUTOMORPHIC-FORMS AND LORENTZIAN KAC-MOODY ALGEBRAS - PART I

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and paraboli c type, we classify all symmetric (and twisted to symmetric) hyperboli c generalized Cartan matrices of elliptic type and of rank 3 with a la ttice Weyl vector. We develop the general theory of reflective lattice s T with 2 negative squares and reflective automorphic forms on homoge neous domains of type IV defined by T. We consider this theory as mirr or symmetric to the theory of elliptic and parabolic hyperbolic reflec tion groups and corresponding hyperbolic root systems. We formulate Ar ithmetic Mirror Symmetry Conjecture relating both these theories and p rove some statements to support this Conjecture. This subject is conne cted with automorphic correction of Lorentzian Kac-Moody algebras. We define Lie reflective automorphic forms which are the most beautiful a utomorphic forms defining automorphic Lorentzian Kac-Moody algebras an d formulate finiteness Conjecture for these forms. Detailed study of a utomorphic correction and Lie reflective automorphic forms for general ized Cartan matrices mentioned above will be given in Part II.