Related papers: Kazhdan-Lusztig conjecture via Zastava spaces

We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the…

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…

We investigate blocks of the Category $\mathcal O$ for the Virasoro algebra over the complex numbers. We demonstrate that the blocks have Kazhdan-Lusztig theories, and that the truncated blocks give rise to interesting Koszul algebras. The…

Graded Hecke algebras can be constructed in terms of equivariant cohomology and constructible sheaves on nilpotent cones. In earlier work, their standard modules and their irreducible modules where realized with such geometric methods. We…

We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Mili\v{c}i\'c-Soergel equivalence of a category of Whittaker modules and a category of Harish-Chandra…

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 Kazhdan-Lusztig theories which was…

Let $\mathfrak{g}$ be a simple, simply-laced Lie algebra and $f \in \mathfrak{g}$ nilpotent. The Kazhdan-Lusztig category of the W-algebra $W^\kappa(\mathfrak{g},f)$ associated with $(\mathfrak{g},f)$ at level $\kappa \in \mathbb{C}$ is…

We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted D-modules on the associated flag variety using Beilinson-Bernstein localization. The main result of this paper is the…

The main result in this paper is the character formula for arbitrary irreducible highest weight modules of W algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, that constructs the W algebras from affine…

We compute projective dimension of translated simple modules in the regular block of the BGG category $\mathcal{O}$ in terms of Kazhdan-Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a…

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 consequence, the characters of finite-dimensional…

We show that the structure of blocks outside the critical hyperplanes of category O over any symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules by…

We introduce the Double leaves basis, a combinatorial basis for the Hom spaces between two Bott-Samelson-Soergel bimodules. As an application we give a combinatorial algorithm to find, for any given Weyl or affine Weyl group, the set of…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We give a description of characters of irreducible Whittaker modules for $\mathfrak{g}$ with any infinitesimal character, along with a Kazhdan-Lusztig algorithm for computing them.…

In this paper we prove the Kazhdan-Lusztig type character formula for irreducible highest weight modules with positive rational highest weights over symmetrizable Kac-Moody Lie algebras.

We prove Conjecture 5.7 in [arXiv:1409.2532], describing all inclusions between primitive ideals for the general linear superalgebra in terms of the Ext1-quiver of simple highest weight modules. For arbitrary basic classical Lie…

In this paper, we show Kazhdan-Lusztig categories, that is, the categories of lower bounded generalized weight modules for certain affine vertex operator superalgebras that are locally finite modules of the underlying finite dimensional Lie…

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking.…

We show that Lusztig's conjecture on the irreducible characters of a reductive algebraic group over a field of positive characteristic is equivalent to the generic multiplicity conjecture, which gives a formula for the Jordan-H"older…

Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in…