The brundan-kazhdan-lusztig conjecture for general linear lie superalgebras

@article{Cheng2015TheBC,
  title={The brundan-kazhdan-lusztig conjecture for general linear lie superalgebras},
  author={Shun-Jen Cheng and Ngau Lam and Weiqiang Wang},
  journal={Duke Mathematical Journal},
  year={2015},
  volume={164},
  pages={617-695}
}
In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan's conjecture and its variants associated to all Borel subalgebras in full generality. 
Representations of the general linear Lie superalgebra in the BGG category O
This is a survey of some recent developments in the highest weight repesentation theory of the general linear Lie superalgebra gln|m(C). The main focus is on the analog of the Kazhdan-Lusztig
GRADED SUPER DUALITY FOR GENERAL LINEAR LIE SUPERALGEBRAS
We provide a new proof of the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras conjectured by Cheng and Wang and first proved by Cheng
Kazhdan-Lusztig Theory of super type D and quantum symmetric pairs
We reformulate the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang
Jantzen filtration and strong linkage principle for modular Lie superalgebras
In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in
Whittaker Modules for Classical Lie Superalgebras
We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Miličić-Soergel equivalence of a category of Whittaker modules
Schur-Weyl duality and categorification
In some joint work with Kleshchev in 2008, we discovered a higher level analog of Schur-Weyl duality, relating parabolic category O for the general linear Lie algebra to certain cyclotomic Hecke
Tilting modules for classical Lie superalgebras
We study tilting and projective‐injective modules in a parabolic BGG category O for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie
On semisimplicity of Jantzen middles for the periplectic Lie superalgebra
We prove that an integral block of the category O of the periplectic Lie superalgebra contains a non-semisimple Jantzen middle if and only if it contains a simple module of atypical highest weight.
A New Approach to Kazhdan-lusztig Theory of Type $b$ Via Quantum Symmetric Pairs
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
...
...

References

SHOWING 1-10 OF 33 REFERENCES
Super duality and Kazhdan-Lusztig polynomials
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a
Brundan-Kazhdan-Lusztig and super duality conjectures
We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their
Super duality for general linear Lie superalgebras and applications
We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of
Irreducible Characters of General Linear Superalgebra and Super Duality
We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating
Tilting Modules for Lie Superalgebras
Abstract We develop a general theory of tilting modules for graded Lie superalgebras, extending work of Soergel for graded Lie algebras. The main result of the article gives a twisted version of BGG
Super duality and irreducible characters of ortho-symplectic Lie superalgebras
We formulate and establish a super duality which connects parabolic categories O for the ortho-symplectic Lie superalgebras and classical Lie algebras of BCD types. This provides a complete and
Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function
Abstract:We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level 1 case. The analysis of this
Categorification of integrable representations of quantum groups
We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig’s
...
...