Related papers: On Singular Localization of $\mathfrak{g}$-modules

In this paper, we treat $\mathscr{D}$-modules on the basic affine space $G/U$ and their global sections for a semisimple complex algebraic group $G$. Our aim is to prepare basic results about large non-irreducible modules for the branching…

Let $\mathfrak g$ be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra {\bf W}$(n)$. We study weight $\mathfrak g$-modules using a method inspired by Mathieu's classification of the simple weight modules with finite…

We prove that, for any fields $k$ and $\mathbb{F}$ of characteristic $0$ and any finite group $T$, the category of modules over the shifted Green biset functor $(kR_{\mathbb{F}})_T$ is semisimple.

This is the second paper of a series of papers on a version of categories $\mathcal{O}$ for root-reductive Lie algebras. Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic…

Let $\mathfrak g$ be a simple complex Lie algebra. In this paper we study the BGG category $\mathcal O_q$ for the quantum group $U_q(\mathfrak g)$ with $q$ being a root of unity in a field $K$ of characteristic $p >0$. We first consider the…

This article provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed by W.Schmid in [Sch]. A corresponding problem in the compact group setting was solved by…

We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique…

We prove that the tensor product of a simple and a finite dimensional $\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $\mathfrak{q}(n)$-supermodules to that of simple…

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to…

We introduce a (bi)category $\mathfrak{Sing}$ whose objects can be functorially assigned spaces of distributions and generalized functions. In addition, these spaces of distributions and generalized functions possess intrinsic notions of…

We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This reduces the problem of classification of simple Virasoro modules which are locally finite over a…

In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of…

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the…

We study the Whittaker category $\mathcal N(\zeta)$ of the Lie superalgebra $\mathfrak g$ for an arbitrary character $\zeta$ of the even subalgebra of the nilpotent radical associated with a triangular decomposition of $\mathfrak g$. We…

We show that the category of finite-dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class…

The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite…

We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories…

We describe the applications of localization methods, in particular the functorial localization formula, in the proofs of several conjectures from string theory. Functorial localization formula pushes the computations on complicated moduli…

We study the categories of singularities coming from Landau-Ginzburg models given by the invertible polynomials. Such categories appear on the B-side of the Berglund-H\"ubsch mirror symmetry. We provide an efficient method of computing…