Generalized Weyl modules, alcove paths and Macdonald polynomials

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr–Shimozono formula in order to prove that the t= ∞ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.