Valery Gritsenko, Université Lille and HSE
Title: Kac-Moody algebras, Borcherds products and L-functions
Abstract: Recently we found with V. Nikulin a new class of Lorentzian Каc-Moody algebras with complete 2-reflective Weyl groups and 2-reflective automorphic denominator functions. These sporadic algebras, the automorphic corrections of hyperbolic Kac-Moody algebras, reflect one of the the largest symmetry structure of the Universe and appear in different physical models. In this talk, I describe briefly the new 2-reflective class of algebras whose classification is based on the arithmetic theory of hyperbolic lattices, the corresponding Borcherds products and some applications. In particular, I present
1) The main reflective towers of the automorphic Borcherds products of the 2-reflective class which are (new) eigenfunctions of all Hecke operators, i.e. they define automorphic representations of O(2,n);
2) The first example of two very different automorphic corrections of the same hyperbolic Kac-Moody algebra;
3) An infinite series of Siegel cusp forms of weight 3 with Borcherds product constructed by theta-quarks and based on a canonical differential form on a modular Calabi-Yau three-fold (the Barth-Nieto quintic).