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Automorphic forms and Lorentzian Kac-Moody algebras. I. (English) Zbl 0935.11015

This paper is concerned with the generalized theory of reflective automorphic forms related to the mirror symmetry and Lorentzian Kac-Moody algebras. In particular, the authors develop the general theory of reflective automorphic forms on Hermitian symmetric domains of type IV associated to reflective lattices with two negative squares, which is viewed as mirror symmetric to the theory of elliptic and parabolic hyperbolic groups and corresponding hyperbolic root systems. They also propose two conjectures and prove some results supporting the conjectures. One is the arithmetic mirror symmetry conjecture which relates the two theories described above. The other is the finiteness conjecture for Lie reflective automorphic forms, which is closely linked to finiteness results for hyperbolic generalized Cartan matrices of elliptic and parabolic type with a lattice Weyl vector. As a special case of such results, the authors classify all symmetric hyperbolic generalized Cartan matrices of elliptic type of rank three with a lattice Weyl vector.
For Part II, see Zbl 0935.11016 below.

MSC:

11F22 Relationship to Lie algebras and finite simple groups
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F50 Jacobi forms
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)

Citations:

Zbl 0935.11016
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