A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs

Abstract

We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair. A new theory of canonical bases arising from quantum symmetric pairs is initiated. It is then applied to formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$. In particular, our approach provides a new formulation of the Kazhdan-Lusztig theory for Lie algebras of type B/C.

v3, book format, with labels of chapters/theorems/equations compatible with the published version

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• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• Tensor product categorifications and the super Kazhdan-Lusztig conjecture
• Crystal structures arising from representations of $GL(m|n)$
• Geometric Schur duality of classical type
• Kazhdan-Lusztig Theory of super type D and quantum symmetric pairs

Cited by in corpus (67)

• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• The bar involution for quantum symmetric pairs
• Semi-infinite highest weight categories
• Universal K-matrices for quantum Kac-Moody algebras
• Categorification of quantum symmetric pairs I
• Whittaker modules for classical Lie superalgebras
• Braided module categories via quantum symmetric pairs
• Hall algebras and quantum symmetric pairs I: foundations
• Canonical bases arising from quantum symmetric pairs of Kac-Moody type
• Geometric Schur duality of classical type
• A Drinfeld type presentation of affine $ı$quantum groups II: split BCFG type
• Affine flag varieties and quantum symmetric pairs
• A Serre presentation for the $ı$quantum groups
• Positivity of Canonical Bases under Comultiplication
• Representations of twisted Yangians of types B, C, D: II
• $ı$Hall algebra of the projective line and $q$-Onsager algebra
• A Fock space model for decomposition numbers for quantum groups at roots of unity
• The $q$-Schur algebras and $q$-Schur dualities of finite type
• A Drinfeld type presentation of affine $ı$quantum groups I: split ADE type
• Character formulae in category $\mathcal{O}$ for exceptional Lie superalgebras $D(2|1;\zeta )$
• Homological invariants in category O for the general linear superalgebra
• Positivity vs negativity of canonical bases
• Formulae of $ı$-divided powers in ${\mathbf{U}}_q({\mathfrak{sl}}_2)$
• Kac-Wakimoto character formula for ortho-symplectic Lie superalgebras
• Schur algebras and quantum symmetric pairs with unequal parameters
• Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals
• Canonical bases arising from quantum symmetric pairs
• Lectures on dualities ABC in representation theory
• On $q$-Schur algebras corresponding to Hecke algebras of type B
• Based modules over the $ı$quantum group of type AI
• Defining relations for quantum symmetric pair coideals of Kac-Moody type
• Formulae of $ı$-divided powers in ${\mathbf{U}}_q({\mathfrak{sl}}_2)$, II
• Quiver varieties and symmetric pairs
• Braid group symmetries on quasi-split $ı$quantum groups via $ı$Hall algebras
• Parabolic category ${\mathcal{O}}^{\mathfrak{p}}$ for periplectic Lie superalgebras $\mathfrak{pe}(n)$
• Global crystal bases for integrable modules over a quantum symmetric pair of type AIII
• Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras
• Canonical bases of a coideal subalgebra in $U_q({\mathfrak{sl}}_2)$
• Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras
• Reflective centers of module categories and quantum K-matrices
• $ı$Hall algebras of weighted projective lines and quantum symmetric pairs
• Geometric Schur Duality of two parameter quantum group of type A
• Cells in affine q-Schur algebras
• A Serre presentation for the $ı$quantum covering groups
• Quasi-solvable lattice models for ${\mathrm{Sp}}_{2n}$ and ${\mathrm{SO}}_{2n+1}$ Demazure atoms and characters
• $ı$Hall algebra of Jordan quiver and $ı$Hall-Littlewood functions
• $ı$Schur duality and Kazhdan-Lusztig basis expanded
• On canonical bases of Letzter algebra ${\mathbf{U}}^{ı}({\mathfrak{sl}}_2)$
• Quantum Schur duality of affine type C with three parameters
• Projective modules over classical Lie algebras of infinite rank in the parabolic category
• Crystal bases of modified $ı$quantum groups of certain quasi-split types
• Multiparameter quantum Schur duality of type B
• Blocks and characters of $G(3)$-modules of non-integral weights
• Characters for Projective Modules in the BGG Category $\mathcal{O}$ for the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(3|4)$
• The i-quantum group U^i(n)
• Character formulae in category $\mathcal{O}$ for exceptional Lie superalgebra $G(3)$
• Stokes phenomenon and reflection equations
• Freidel-Maillet type presentations of $U_q(s{l}_2)$
• Canonical bases arising from $ı$quantum covering groups
• Vust's theorem and higher level Schur-Weyl duality for types B, C and D
• Positivity of $ı$-canonical bases
• A new realization of the i-quantum group U^j(n)
• Quasi-split symmetric pairs of $U({\mathfrak{gl}}_N)$ and their Schur algebras
• PBW bases for modified quantum groups
• G(l,k,d)-modules via groupoids
• Geometric Howe dualities of finite type
• $ı$quantum groups of split type via derived Hall algebras