The Deformed Tanisaki-Garsia-Procesi Modules

The polynomial ideals studied by A. Garsia and C. Procesi play an important role in the theory of Kostka polynomials. We give multiparameter flat deformations of these ideals and define an action of the extended affine symmetric group on the corresponding quotient algebras multiplied by the sign representation. We show that the images of these modules under the affine Schur-Weyl duality are dual to the local Weyl modules for the loop algebra \(\mathfrak {sl}_{n+1}[t^{\pm 1}]\) ${\mathfrak{sl}}_{n+1}[t^{\pm 1}]$ .