Alcove walks: from Macdonald polynomials to quantum K-theory

I will present some recent applications of the alcove model in the representation theory of semisimple Lie algebras and, more generally, of symmetrizable Kac-Moody algebras. This is a discrete counterpart of the Littelmann path model, which was defined in my joint work with A.
Postnikov. The first part of the talk is concerned with applications to Macdonald polynomials, which are generalizations of the irreducible characters of semisimple Lie algebras depending on two parameters. Then I will mention an application to a Chevalley-type multiplication formula in the K-theory of generalized flag varieties G/B, as well as to a conjectured formula of the same type in quantum K-theory.