有限域上辛向量空间的量子化

Quantizations of conical symplectic resolutions II: category $\mathcal O$ and symplectic duality

We define and study category $\mathcal O$ for a symplectic resolution, generalizing the classical BGG category $\mathcal O$, which is associated with the Springer resolution. This includes the development of intrinsic properties parallelling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of individual examples. We observe that category $\mathcal O$ is often Koszul, and its Koszul dual is often equivalent to category $\mathcal O$ for a different symplectic resolution. This leads us to define the notion of a s…

特殊辛幂零轨道的量子化及其闭包的正规性

Quantization of Special Symplectic Nilpotent Orbits and Normality of their Closures

本文研究了正则函数环$R(\mathcal{O})$对所有具有偶数列长的辛幂零轨道$\mathcal{O}$的作用.我们首先回顾量子化模型的所有这些轨道的Barbasch使用幂幺表示。在这个模型中，我们用一个抛物线诱导模来表示出现在$R(\mathcal{O})$中的基本表示的多重性。最后，我们将利用这个公式给出$\mathcal{O}$的Zagliki闭包$\bar{\mathcal{O}}$的正规性的一个判据。

The canonical global quantization of symplectic varieties in characteristic $p$

Let $X$ be a smooth symplectic variety over a field $k$ of characteristic $p>2$ equipped with a restricted structure, which is a class $[η] \in H^0(X, Ω^1_X/d\mathcal O_X)$ whose de Rham differential equals the symplectic form. In this paper we construct a functorial in $(X, [η])$ formal quantization of the category $\mathrm{QCoh}(X)$ of quasi-coherent sheaves on $X$. We also construct its natural extension to a quasi-coherent sheaf of categories $\mathrm{QCoh}_h$ on the product $X^{(1)} \times {\mathbb S}$ of the Frobenius twist of $X$ and the projective line ${\mathbb S}=\mathbb P^1$, viewed…

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…

辛球面幂零轨道的正则函数及其量子化

Regular Functions of Symplectic Spherical Nilpotent Orbits and their Quantizations

研究了$G = Sp(2n,\mathbb{C})$的经典球轨道$R(\mathcal{O})$的正则函数环.特别地，把$G$看作是一个具有极大紧子群$K$的实李群，当$\mathcal{O}$是幂零轨道$(2^{2p}1^{2q})$时，我们研究了$\mathcal{O}$的量子化模型.利用这个模型，我们验证了McGovern的一个猜想和Achar和Sommers的另一个猜想。假设[Barbasch 2008]中的结果，我们还将验证更大类幂零轨道的Achar-Sommers猜想。

Deformations of symplectic singularities and Orbit method for semisimple Lie algebras

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of adjoint orbits in a semisimple Lie algebra to the set of primitive ideals in the universal enveloping algebra. We show that the map is in…

Derived equivalences for Symplectic reflection algebras

In this paper we study derived equivalences for Symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over Symplectic reflection algebras and categories of coherent sheaves over quantizations of Q-factorial terminalizations of the symplectic quotient singularities. To do this we construct a Procesi sheaf on the terminalization and show that the quantizations of the terminalization are simple sheaves of algebras. We will also sketch some applications: to the generalized Bernstein inequality and to perversity of wall crossing fun…

量子化辛分解的跨壁函子：反常性和部分Ringel对偶

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

本文研究了辛分解量子化上模范畴之间的跨壁函子。我们证明，穿墙函子通过面对反常的等价，并使用此验证Etingof型猜想的量化Nakajima的变种与仿射箭。在这种情况下，当有一个哈密顿环面的决议与许多不动点，使它有意义的范畴$\mathcal{O}$在量子化的发言，我们引入新的标准分层结构，这些类别$\mathcal{O}$和相关的Ringel对偶函子与这些标准分层结构的作用。

Harish-Chandra bimodules over quantized symplectic singularities

In this paper we classify the irreducible Harish-Chandra bimodules with full support over filtered quantizations of conical symplectic singularities under the condition that none of the slices to codimension 2 symplectic leaves has type $E_8$. More precisely, we show that the top quotient $\overline{\operatorname{HC}}(\mathcal{A}_λ)$ of the category of Harish-Chandra bimodules over the quantization $\mathcal{A}_λ$ with parameter $λ$ embeds into the category of representations of the algebraic fundamental group, $Γ$, of the open leaf. The image coincides with the representations of $Γ/Γ_λ$, whe…

几何量子化与酉最高权Harish-Chandra超模

Geometric quantization and unitary highest weight Harish-Chandra supermodules

本文把几何量子化推广到超设置，对逆步李超群的实形式在伪Kähler超流形上的作用构造酉表示，得到最高权Harish-Chandra超模并建立Gelfand模型，且证明量子化与辛约化可交换。