Generalized Weyl modules, alcove paths and Macdonald polynomials

TL;DR

This paper extends the concept of local Weyl modules to arbitrary weights, linking their representation theory to alcove paths and Macdonald polynomials, and providing new character formulas.

Contribution

It introduces a generalized definition of Weyl modules for arbitrary weights and connects their characters to alcove paths and Macdonald polynomials using the Orr-Shimozono formula.

Findings

Generalized Weyl modules are characterized by alcove paths and quantum Bruhat graph.

The $t=\infty$ specializations of nonsymmetric Macdonald polynomials match the characters of these modules.

The work broadens the understanding of Weyl modules beyond dominant weights.

Abstract

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.