Representations with minimal support for quantized Gieseker varieties

Etingof conjecture for quantized quiver varieties

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof's conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categor…

The bar automorphism in quantum groups and geometry of quiver representations

Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.

Quantum flag varieties, equivariant quantum D-modules and localization of quantum groups

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 $\Oq(B)$-equivariant $\Oq(G)$-modules and proves that this is a proj-category. We construct a category of equivariant quantum $\mathcal{D}$-modules on this quantized flag variety and prove the Beilinson-Bernsteins localization theorem for this category in the case when $q$ is not a root of unity.

$K$-理论的有向图规范理论库仑分支与聚类流形

$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

本文研究了有向图规范理论库仑分支与聚类流形之间的关系，证明了量化K-理论库仑分支代数与上聚类X-代数的同构性，并展示了初始种子的聚类Donaldson-Thomas变换。

带关系箭图的通用 Virasoro 约束

Universal Virasoro constraints for quivers with relations

本文提出箭图表示模空间 Virasoro 约束的通用方法，证明其对任意带关系、含冻结顶点且无环的有限箭图成立，并借助导出等价给出射影平面及双射影线上 Gieseker 半稳定层 Virasoro 约束的独立证明，进而独立证明任意曲面上秩 1 情形。

Homological description of crystal structures on Lusztig's quiver varieties

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions ar…

Wall-crossing functors for quantized symplectic resolutions: perversity and partial Ringel dualities

In this paper we study wall-crossing functors between categories of modules over quantizations of symplectic resolutions. We prove that wall-crossing functors through faces are perverse equivalences and use this to verify an Etingof type conjecture for quantizations of Nakajima quiver varieties associated to affine quivers. In the case when there is a Hamiltonian torus action on the resolution with finitely many fixed points so that it makes sense to speak about categories $\mathcal{O}$ over quantizations, we introduce new standardly stratified structures on these categories $\mathcal{O}$ and …

On injective modules and support varieties for the small quantum group

Let $u_ζ(g)$ denote the small quantum group associated to the simple complex Lie algebra $g$, with parameter $q$ specialized to a primitive $\ell$-th root of unity $ζ$ in the field $k$. Generalizing a result of Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional $u_ζ(g)$-module admitting a compatible torus action, then the injectivity of $M$ as a module for $u_ζ(g)$ can be detected by the restriction of $M$ to certain root subalgebras of $u_ζ(g)$. If the characteristic of $k$ is positive, then this injectivity criterion also holds for the higher Frobenius--Lusztig kernels $U…

Quantizations of conical symplectic resolutions I: local and global structure

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing t…

Berezin quantization of noncommutative projective varieties

We use operator algebras and operator theory to obtain new result concerning Berezin quantization of compact Kähler manifolds. Our main tool is the notion of subproduct systems of finite-dimensional Hilbert spaces, which enables all involved objects, such as the Toeplitz operators, to be very conveniently expressed in terms of shift operators compressed to a subspace of full Fock space. This subspace is not required to be contained in the symmetric Fock space, so from finite-dimensional matrix algebras we can construct noncommutative manifolds with extra structure generalizing that of a projec…