Related papers: Cohomology of Standard Modules on Partial Flag Var…

We give a geometric realization of cohomologically induced (g,K)-modules. Let (h,L) be a subpair of (g,K). The cohomological induction is an algebraic construction of (g,K)-modules from a (h,L)-module V. For a real semisimple Lie group, the…

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomolgy of an equivariant holomorphic line bundle defined on an open invariant submanifold of a…

Let $G$ be a simple, simply-connected complex algebraic group with Lie algebra $\mathfrak{g}$, and $G/B$ the associated complete flag variety. The Hochschild cohomology $HH^\bullet(G/B)$ is a geometric invariant of the flag variety related…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. The Beilinson-Bernstein localization theorem establishes an equivalence of the category of $\mathfrak{g}$-modules of a fixed infinitesimal character and a category of modules over a…

Beilinson-Bernstein localization realizes representations of complex reductive Lie algebras as monodromic $D$-modules on the "basic affine space" $G/N$, a torus bundle over the flag variety. A doubled version of the same space appears as…

We study a variety of questions centered around the computation of cohomology of line bundles on the incidence correspondence (the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane…

For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting…

Beilinson--Bernstein localisation relates representations of a Lie algebra $\mathfrak{g}$ to certain $\mathcal{D}$-modules on the flag variety of $\mathfrak{g}$. In [arXiv:2002.01540], examples of $\mathfrak{sl}_2$-representations which…

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie…

In this manuscript we make a general study of the representations realized, for a reductive Lie group of Harish-Chandra class, on the compactly supported sheaf cohomology groups of an irreducible finite-rank polarized homogeneous vector…

Let $\Oq(G)$ be the algebra of quantized functions on an algebraic group $G$ and $\Oq(B)$ its quotient algebra corresponding to a Borel subgroup $B$ of $G$. We define the category of sheaves on the "quantum flag variety of $G$" to be the…

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These…

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing…

We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof…

A relationship between quantum flag and Grassmann manifolds is revealed. This enables a formal diagonalization of quantum positive matrices. The requirement that this diagonalization defines a homomorphism leads to a left \Uh -- module…

Vinberg--Kimel'fel'd [Funct. Anal. Appl., 1978] established that a quasi-projective normal $G$-variety $X$ is spherical if and only if $G$-modules on the spaces $\Gamma(X, \mathcal{L})$ of global sections of $G$-equivariant line bundles are…

Regular semisimple Hessenberg varieties admit actions of associated Weyl groups on their cohomology space of each degree. In this paper, we consider the module structure of the cohomology spaces of regular semisimple Hessenberg varieties of…

We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an…

We formulate a Beilinson-Bernstein type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping…

A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially…