Related papers: The half-quantum flag variety and representations …

De Concini, Kac, and Procesi defined a family of subalgebras Uq[w] of the quantized enveloping algebra Uq(g) associated to elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We…

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the…

Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a…

For each compact, simple, simply-connected Lie group and each integer level we construct a modular tensor category from a quotient of a certain subcategory of the category of representations of the corresponding quantum group. We determine…

We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type $A_1^{(1)}$ to the category of equivariant perverse coherent sheaves on the nilpotent cone of type $A$. We prove…

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of…

In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra $\mathfrak{g}$. This problem reduces to the classification of all Lie bialgebra structures on…

The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a…

The category $\cal{C}$ (studied by Andersen-Jantzen-Soergel) of representations of a Lusztig's small quantum group at a root of unity, together with its modular structure, is defined geometrically, using configuration spaces.

This thesis provides a partial answer to a question posed by Greg Kuperberg in q-alg/9712003 and again by Justin Roberts as problem 12.18 in "Problems on invariants of knots and 3-manifolds", math.GT/0406190, essentially: "Can one describe…

Let G be an infinitesimal group scheme of finite height r and V(G) the scheme which represents 1-parameter subgroups of G. We consider sheaves over the projectivization P(G) of V(G) constructed from a G-module M. We show that if P(G) is…

Let $X$ be a smooth scheme with an action of an algebraic group $G$. We establish an equivalence of two categories related to the corresponding moment map $\mu : T^*X \to Lie(G)^*$ - the derived category of G-equivariant coherent sheaves on…

We study multiplicative properties of the (quantum) dual canonical basis B* associated to a semi-simple complex Lie group G. We provide a subset D of B* such that the following property holds : if two elements b, b' in B* q-commute and if…

We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same…

Let $X$ be a smooth projective variety acted on by a reductive group $G$. Let $L$ be a positive $G$-equivariant line bundle over $X$. We use the Witten deformation of the Dolbeault complex of $L$ to show, that the cohomology of the sheaf of…

Let X be e quasi-compact and semi-separated scheme. If every at quasi- coherent sheaf has finite cotorsion dimension, we prove that X is n-perfect for some n > 0. If X is coherent and n-perfect(not necessarily of finite krull dimension), we…

Let G be a semisimple complex Lie group. In this article, we study Geometric Invariant Theory on a flag variety G/B with respect to the action of a principal 3-dimensional simple subgroup S of G. We determine explicitly the GIT-equivalence…

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr$_{GL(n)}$) of (all) D-modules on the affine Grassmannian of the group $GL(n)$ as the category of quasi-coherent sheaves on a certain stack (it is…

We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new…

In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit…