The 2-parameter Green functions for 8-dimensional spin groups

素幂阶循环群的Green环上的Adams运算

Adams operations on the Green ring of a cyclic group of prime-power order

本文研究了素特征$p$的域$K$上循环$p$群$C$的Green环$R_{KC}$，并确定了当$n$不能被$p$整除时的Adams运算$ψ^n$.这给出了分解成$KC$ -模的外幂和对称幂的不可分解的信息。

Formulae for two-variable Green functions

Based on results of Digne-Michel-Lehrer (2003) we give two formulae for two-variable Green functions attached to Lusztig induction in a finite reductive group. We present applications to explicit computation of these Green functions, to conjectures of Malle and Rotilio, and to scalar products between Lusztig inductions of Gelfand-Graev characters.

有限约化群的广义格林函数与幂幺类，I

Generalized Green functions and unipotent classes for finite reductive groups, I

计算有限约化群的广义格林函数的算法中含有一些未知标量，这些标量来自G上幂单类上的不可约局部系统的F_q结构。本文在G = SL_n且F是分裂型或非分裂型的情况下，确定了这类标量。在F是非分裂型的情况下，我们利用Lusztig的分次Hecke代数及其在等变同调群上的表示理论.

Generalized Green functions associated to complex reflection groups

In this paper, we consider the set of r-symbols in a full generality. We construct Hall-Littlewood functions and Kostka functions associated to those r-symbols. We also discuss a multi-parameter version of those functions. We show that there exists a general algorithm of computing multi-parameter Kostka functions. As an application, we show that the generalized Green functions of symplectic groups can be described combinatorially in terms of those (one-parameter) Kostka functions.

Linear source invertible bimodules and Green correspondence

We show that the Green correspondence induces an injective group homomorphism from the linear source Picard group $\mathcal{L}(B)$ of a block $B$ of a finite group algebra to the linear source Picard group $\mathcal{L}(C)$, where $C$ is the Brauer correspondent of $B$. This homomorphism maps the trivial source Picard group $\mathcal{T}(B)$ to the trivial source Picard group $\mathcal{T}(C)$. We show further that the endopermutation source Picard group $\mathcal{E}(B)$ is bounded in terms of the defect groups of $B$ and that when $B$ has a normal defect group $\mathcal{E}(B)=\mathcal{L}(B)$. Fi…

Ringel-Hall algebras beyond their quantum groups I: Restriction functor and Green's formula

In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.

有限群的Green环上Adams运算的周期性

Periodicity of Adams operations on the Green ring of a finite group

域$K$上的群$G$的Green环上的Adams运算$ψ_Λ^n$和$ψ_S^n$为研究$KG$ -模的外幂和对称幂提供了一个框架.当$G$是有限的且$K$具有素特征$p$时，我们证明了$ψ_Λ^n$和$ψ_S^n$在$n$中是周期的当且仅当$G$的Sylow $p$ -子群是循环的.在$G$是一个循环的$p$ -组的情况下，我们找到的最小周期，并使用西蒙兹最近的工作来表达$ψ_S^n$的$ψ_Λ^n$。

有限约化群的广义Green函数与幂幺类，IV

Generalized Green functions and unipotent classes for finite reductive groups, IV

本文定义了连通约化群幂幺类的分裂元，将广义Green函数的Lusztig算法计算归结为分裂元存在性，并证明典型群及除E7型一个类外的例外群中分裂元均存在。

Remarks on computing Green functions

We describe the computation of generalized Green functions and 2-parameter Green functions for finite reductive groups.

外尔群的Green多项式、椭圆对和广义Dirac指数

Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index

我们提供了一个直接的连接之间的Springer理论，通过格林多项式，不可约表示的针覆盖$\wti W$，一定的双重覆盖的外尔群$W$，和一个扩展的狄拉克算子分次Hecke代数。我们的方法导致一个新的和统一的建设不可约真正的$\wti W$ -字符。在这个过程中，我们给出了Springer理论上同调群上Dynkin图的外自同构作用的构造，并引入了$W$关于反射表示$V$的$q$ -椭圆配对.这些建筑是独立的利益。$q$ -椭圆配对是Reeder提出的$W$的椭圆配对的推广，它与S.半直积$A_W=\bC[W]\ltimes S(V)$的Kato（分次）Kostka系统的概念.

Picture群与极大绿色序列

Picture groups and maximal green sequences

本文证明Picture群与有限型带值Dynkin箭图的极大绿色序列存在双射，并将结果推广至任意遗传代数，同时刻画了picture的原子分解。

张量积与对称群代数的稳定绿色环

Tensor Product and the Stable Green Ring of the Symmetric Group Algebra $F\mathfrak{S}_p$

研究对称群代数F S_p的不可约非投射模的张量积分解，证明其模投射模为半单，并计算Benson-Symonds不变量。

Green functions and Glauberman degree-divisibility

The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality.

小特征下的Green函数

Green Functions in Small Characteristic

本文解决了Lie型群普通Green函数最后未决情形，即当q为2、3或5的幂时例外群E_8(q)的Green函数值。

从Ringel–Hall代数的视角看Picture群与Green序列

Picture groups and green sequences from the perspective of Ringel--Hall algebras

该研究从Ringel–Hall代数视角，为有限维结合k-代数Λ构造忠实群函子，证明其Picture空间局部CAT(0)的条件，还将Green序列与Picture群正表达式的一一对应结果推广到非遗传k-代数。

基于双曲局部化的格林函数

Green functions via hyperbolic localization

设G是一个约化代数群，具有幂零锥N和标志簇G/B。本文构造了一个从N上的反常层到G/B上的局部常数层的正合函子，并利用它研究了N上的单反常层的Ext群.作为应用，我们给出了N上反常层的茎的一些已知结果的新证明。