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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]

The characters of the (total) Springer representations afford the Green functions, that can understood as generalizations of Hall–Littlewood's Q -functions. In this paper, we present a purely algebraic proof that the (total) Springer representations of GL (n) are Ext -orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions. [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]

Abstract: We study the orbits of in the enhanced nilpotent cone , where is the variety of nilpotent endomorphisms of V. These orbits are parametrized by bipartitions of , and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato''s exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled. [Copyright &y& Elsevier]

Abstract: We construct a Poincaré–Birkhoff–Witt type basis for the Weyl modules [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191–223, math.QA/0004174] of the current algebra of . As a corollary we prove the conjecture made in [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191–223, math.QA/0004174; V. Chari, A. Pressley, Integrable and Weyl modules for quantum affine , in: Quantum Groups and Lie Theory, Durham, 1999, in: London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 48–62, math.QA/0007123] on the dimension of the Weyl modules in this case. Further, we relate the Weyl modules to the fusion modules defined in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimensional Lie Algebras, and Applications, in: Amer. Math. Soc. Transl. Ser. 2, vol. 194, 1999, pp. 61–79, math.QA/9812093] of the current algebra and the Demazure modules in level one representations of the corresponding affine algebra. In particular, this allows us to establish substantial cases of the conjectures in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimensional Lie Algebras, and Applications, in: Amer. Math. Soc. Transl. Ser. 2, vol. 194, 1999, pp. 61–79, math.QA/9812093] on the structure and graded character of the fusion modules. [Copyright &y& Elsevier]

Abstract: A new fermionic formula for the unrestricted Kostka polynomials of type is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov–Reshetikhin modules, not just for the symmetric and antisymmetric case. The fermionic formula can be interpreted in terms of a new set of unrestricted rigged configurations. For the proof a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths is given which generalizes a bijection of Kirillov and Reshetikhin. [Copyright &y& Elsevier]

Abstract: Creation operators are given for the three distinguished bases of the type BCD universal character ring of Koike and Terada yielding an elegant way of treating computations for all three types in a unified manner. Deformed versions of these operators create symmetric function bases whose expansion in the universal character basis, has polynomial coefficients in q with nonnegative integer coefficients. We conjecture that these polynomials are one-dimensional sums associated with crystal bases of finite-dimensional modules over quantized affine algebras for all nonexceptional affine types. These polynomials satisfy a Macdonald-type duality. [Copyright &y& Elsevier]