Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems

We define the formal affine Demazure algebra and a formal affine Hecke algebra associated to a Kac-Moody root system. We prove structure theorems for these algebras, and use them to extend several results and constructions (presentation in terms of generators and relations, coproduct and product structures, filtration by codimension of Bott-Samelson classes, root polynomials and multiplication formulas) that were previously known for finite root systems.