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Register a new userCategorification of DAHA and Macdonald polynomials
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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.
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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.
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.
Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type $mathfrak{sl}_2$ at prime roots of unity by Elias-Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones.
We study a presentation of Khovanov - Lauda - Rouquiers candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of $2$-cells in this $2$-category. We show that these bases correspond to Khovanov and Laudas conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this $2$-category is a categorification of Lusztigs idempotent and integral quantum group $bf{U}_{q}(mathfrak{g})$ associated to a symmetrizable simply-laced Kac-Moody algebra $mathfrak{g}$.
Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. In this paper, we study the monodromy of these eigenvectors as the parameter varies within the real locus; this gives an action of the fundamental group of this moduli space, which is called the cactus group. We prove a conjecture of Etingof which states that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of g-crystals. In fact, we prove that the coboundary category of normal g-crystals can be reconstructed using the coverings of the moduli spaces. Our main tool is the construction of a crystal structure on the set of eigenvectors for shift of argument algebras, another family of commutative algebras which act on any irreducible g-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on g-crystals.
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