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Register a new userQuantum Schur duality of affine type C with three parameters
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We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD.
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For a Dynkin quiver $Q$ of type ADE and a sum $beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserots geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclercs isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $beta$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kims generalized quantum affine Schur-Weyl duality functor. As a by-product, we verify a conjecture by Kang-Kashiwara-Kim on the simpleness of some poles of normalized R-matrices for any quiver $Q$ of type ADE.
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclercs monoidal category $mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(Lmathfrak{g})$ via Nakajimas homomorphism. As an application, we show that Kang-Kashiwara-Kims generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with $Q$ and Hernandez-Leclercs category $mathcal{C}_Q$, assuming the simpleness of some poles of normalized R-matrices for type E.
We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type A III/IV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztigs bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li from a geometric viewpoint.
Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an $imath$Schur duality between an $imath$quantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wangs quasi-split $imath$Schur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $imath$canonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the $imath$Schur duality.
Recently, Lu and Wang formulated a Drinfeld type presentation for $imath$quantum group $widetilde{{mathbf U}}^imath$ arising from quantum symmetric pairs of split affine ADE type. In this paper, we generalize their results by establishing a current presentation for $widetilde{{mathbf U}}^imath$ of arbitrary split affine type.
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