On the cleanness of cuspidal character sheaves

曲线的KLR代数与Schur代数及半尖点表示

KLR and Schur algebras for curves and semi-cuspidal representations

本文在光滑曲线上用挠层替代箭图表示，构造了KLR代数和Schur代数的曲线版本，并证明在$\mathbb P^1$情形下与Kronecker箭图的虚半尖点范畴Morita等价，由此质疑仿射箭图奇偶层理论的合理性。

经典对称对的特征层

Character sheaves for classical symmetric pairs

本研究为经典对称对构建Springer理论，明确刻画该场景下的特征层并确定尖点特征层，推进了相关表示论研究。

分次李代数的不变系统与特征层

Invariant systems and character sheaves for graded Lie algebras

该研究通过引入微分方程不变系统构造分次李代数上的特征层，并与既有结果共同给出Vinberg I型经典情形的全部尖点特征层。

On the computation of character values for finite Chevalley groups of exceptional type

We discuss various computational issues around the problem of determining the character values of finite Chevalley groups, in the framework provided by Lusztig's theory of character sheaves. Some of the remaining open questions (concerning certain roots of unity) for the cuspidal unipotent character sheaves of groups of exceptional type are resolved.

分次李代数的特征层：稳定分次

Character Sheaves for Graded Lie Algebras: Stable Gradings

本研究针对稳定分次李代数，通过分析[GVX2]中邻近周期层的傅里叶变换，构造出推测为尖点特征层的满支撑特征层，涉及单位根处复反射群相关Hecke代数的不可约表示。

对称对的特征层：特殊线性群

Character sheaves for symmetric pairs: special linear groups

本研究利用[VX1]的通用策略与中心特征分析，明确刻画了特殊线性群内对合对应的对称对的特征层，并确定了其中的尖点特征层。

On quadratic distinction of automorphic sheaves

We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ being the base change of a stable cuspidal packet of the quasi-split unitary group associated to the quadratic extension $E/F$, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of $\mathrm{GL}_{n,X'}$ is nonvanishing if and only if the corresponding local system $E$ on $X'$ is conjugate self-dual with respect to an étale double cover $X'/X$ of curves,…

正特性下李代数上的特征层

Character Sheaves on Reductive Lie Algebras in Positive Characteristic

本文在正特性下研究李代数上的特征层，证明了其微局部刻画及可允许性与特征层等价性的几何证明。

On character sheaves and characters of reductive groups at unipotent classes

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front se…

On the values of unipotent characters of finite Chevalley groups of type $E_7$ in characteristic 2

Let $G$ be a finite group of Lie type. In order to determine the character table of $G$, Lusztig developed the theory of character sheaves. In this framework, one has to find the transformation between two bases for the space of class functions on $G$, one of them being the irreducible characters of $G$, the other one consisting of characteristic functions associated to character sheaves. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. The problem of specifying these scalars can be reduced to con…

相对外尔群I，II的作用

Actions of relative Weyl groups I, II

在第一部分中，我们构造了诱导尖点特征标层的自同态代数与相关外尔群的群代数之间的新同构。我们表明，它不同于同构的Lusztig的线性字符，我们将这种线性字符的一些稳定剂。得到了特征层的特征函数的一些结论。在第二部分中，我们明确地计算这个线性字符的尖点局部系统的支持下，由正规的幂幺类，作为这些方法的应用，我们得到了一个改进的Digne，Lehrer和Michel的定理Lusztig限制的凝胶'fand-Graev字符。

Tilting on non-commutative rational projective curves

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded d…

几何 Eisenstein 级数 I：有限性定理

Geometric Eisenstein series I: finiteness theorems

本文在 Fargues–Fontaine 曲线的丛栈上建立几何 Eisenstein 级数与常数项函子理论，证明最优有限性及 Bernstein 第二伴随性的几何形式，并实现 Bun_G 层范畴的尖点–Eisenstein 分解。

纯度与2-Calabi-Yau范畴

Purity and 2-Calabi-Yau categories

本文针对具有良好模空间的2-Calabi-Yau范畴对象栈，利用形式性和étale邻域定理建立混合Hodge模复形及Borel-Moore同调的纯度，进而证明分解定理、定义尖点上同调，并推广到无良好模空间的凝聚层栈及奇异Higgs层栈。