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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 sl n + 1 [ t ± 1 ] . [ABSTRACT FROM AUTHOR]

We study the Q-Kostka polynomials L λ μ (t) by the vertex operator realization of the Q-Hall–Littlewood functions G λ (x ; t) and derive new formulae for L λ μ (t) . In particular, we have established stability property for the Q-Kostka polynomials. We also introduce spin Green polynomials Y μ λ (t) as both an analogue of the Green polynomials and deformation of the spin irreducible characters of S n . Iterative formulas of the spin Green polynomials are given and some favorable properties parallel to the Green polynomials are obtained. Tables of Y μ λ (t) are included for n ≤ 7. [ABSTRACT FROM AUTHOR]

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require. [ABSTRACT FROM AUTHOR]

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules. [ABSTRACT FROM AUTHOR]

We introduce a spin analogue of Kostka polynomials and show that these polynomials enjoy favorable properties parallel to the Kostka polynomials. Further connections of spin Kostka polynomials with representation theory are established. [ABSTRACT FROM AUTHOR]