Reflection Groups in Hyperbolic Spaces and the Denominator Formula for Lorentzian Kac–moody Lie Algebras

This is a continuation of our ”Lecture on Kac–Moody Lie algebras of the arithmetic type” [25]. We consider hyperbolic (i.e. signature (n, 1)) integral symmetric bilinear form S : M × M → Z (i.e. hyperbolic lattice), reflection group W ⊂ W (S), fundamental polyhedron M of W and an acceptable (corresponding to twisting coefficients) set P (M) ⊂ M of vectors orthogonal to faces of M (simple roots). One can construct the corresponding Lorentzian Kac–Moody Lie algebra g = g(A(S,W,P (M))) which is graded by M . We show that g has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if g has so called restricted arithmetic type. We show that every finitely generated (i.e. P (M) is finite) algebra g′′(A(S,W1, P (M1))) may be embedded to g(A(S,W, P (M))) of the restricted arithmetic type. Thus, Lorentzian Kac–Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac–Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have a lattice Weyl vector ρ. Lorentzian Kac–Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector ρ are called elliptic (if S(ρ, ρ) < 0) or parabolic (if S(ρ, ρ) = 0). We use and extend our and Vinberg’s results on reflection groups in hyperbolic spaces to show that the sets of elliptic and parabolic Kac–Moody Lie algebras with generalized lattice Weyl vector and lattice Weyl vector are essentially finite. We also consider connection of these results with the recent results by R.Borcherds.