  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 $t=\infty$ specializations of
the nonsymmetric Macdonald polynomials are equal to the characters of certain
generalized Weyl modules.
