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Register a new userWreath Macdonald polynomials and categorical McKay correspondence
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Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_nltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of ${mathbb A}^{2n}$ by the symmetric group $S_n$. A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin via quantization in positive characteristic. In the present note we show the properties of the derived equivalence which imply the generalized Macdonald positivity for wreath products.
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We describe a categorification of the Double Affine Hecke Algebra ${mathcal{H}kern -.4emmathcal{H}}$ associated with an affine Lie algebra $widehat{mathfrak{g}}$, a categorification of the polynomial representation and a categorification of Macdonald polynomials. All categorification results are given in the derived setting. That is, we consider the derived category associated with graded modules over the Lie superalgera ${mathfrak I}[xi]$, where ${mathfrak I}subsetwidehat{mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $xi$ is a formal odd variable. The Euler characteristic of graded characters of a complex of ${mathfrak I}[xi]$-modules is considered as an element of a polynomial representation. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${mathfrak{p}}_{i}$ categorify Demazure operators $T_i+1in{mathcal{H}kern -.4emmathcal{H}}$, meaning that all algebraic relations of $T_i$ have categorical meanings. Second, we describe a natural collection of complexes ${mathbb{EM}}_{lambda}$ of ${mathfrak I}[xi]$-modules whose Euler characteristic is equal to nonsymmetric Macdonald polynomials $E_lambda$ for dominant $lambda$ and a natural collection of complexes of $mathfrak{g}[z,xi]$-modules ${mathbb{PM}}_{lambda}$ whose Euler characteristic is equal to the symmetric Macdonald polynomial $P_{lambda}$. We illustrate our theory with the example $mathfrak{g}=mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${mathfrak I}[xi]$ such that their supercharacters coincide with renormalizations of nonsymmetric Macdonald polynomials.
A string theoretic derivation is given for the conjecture of Hausel, Letellier, and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with G. Pan. Haimans geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.
We prove a Littlewood-Richardson type formula for $(s_{lambda/mu},s_{ u/kappa})_{t^k,t}$, the pairing of two skew Schur functions in the MacDonald inner product at $q = t^k$ for positive integers $k$. This pairing counts graded decomposition numbers in the representation theory of wreath products of the algebra $C[x]/x^k$ and symmetric groups.
Let H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for H, which conjecturally puts supercuspidal H-representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernsteins theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for H, that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible H-representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for H-representations to the corresponding problem for supercuspidal representations of Levi subgroups of H. The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztigs generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.
In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.
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