Tensor product categorifications and the super Kazhdan-Lusztig conjecture

Abstract

We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie super algebra ${\mathfrak{gl}}_{n|m}(\mathbb{C})$ as formulated originally by the first author. We also prove for the first time that any integral block of category O for ${\mathfrak{gl}}_{n|m}(\mathbb{C})$ (and also all of its parabolic analogs) possesses a graded version which is Koszul. Our approach depends crucially on an application of the uniqueness of tensor product categorifications established recently by the second two authors.

58 pages; v2: relatively minor changes, a few adjustments to wording and references; v3: final version, more minor changes, to appear in IMRN

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• Knot invariants and higher representation theory
• Derived equivalences for symmetric groups and sl\_2-categorification
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Cited by in corpus (27)

• A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs
• Monoidal supercategories
• Rouquier's conjecture and diagrammatic algebra
• Affine highest weight categories and affine quasihereditary algebras
• Semi-infinite highest weight categories
• The primitive spectrum of basic classical Lie superalgebras
• Tilting modules for classical Lie superalgebras
• Integrable $sl(\infty )$-modules and Category $\mathcal{O}$ for $\mathfrak{gl}(m|n)$
• The primitive spectrum for gl(m|n)
• Reduction method for representations of queer Lie superalgebras
• An inversion formula for some Fock spaces
• Homological invariants in category O for the general linear superalgebra
• Categorification of tensor product representations of sl(k) and category O
• Parabolic category ${\mathcal{O}}^{\mathfrak{p}}$ for periplectic Lie superalgebras $\mathfrak{pe}(n)$
• Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras
• Whittaker coinvariants for $\mathrm{GL}(m|n)$
• Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras: A Survey
• Blocks and characters of $G(3)$-modules of non-integral weights
• Essential orders on stratified algebras with duality and $\mathcal{S}$-subcategories in $\mathcal{O}$
• Characters for Projective Modules in the BGG Category $\mathcal{O}$ for the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(3|4)$
• Classification problems in 2-representation theory
• Categorical actions and multiplicities in the Deligne category $\underline{Rep}(G{L}_t)$
• Quantum group of type $A$ and representations of queer Lie superalgebra
• Character formulae in category $\mathcal{O}$ for exceptional Lie superalgebra $G(3)$
• Super duality for quantum affine algebras of type $A$
• On semisimplicity of Jantzen middles for the periplectic Lie superalgebra
• Representation theory of a semisimple extension of the Takiff superalgebra