论文标题
Hecke内态代数的确切类别方法
An exact category approach to Hecke endomorphism algebras
论文作者
论文摘要
令$ g $为有限的谎言类型。在研究$ g $的跨特征代表理论时，（专业的）Hecke代数$ h = \ end_g（\ ind_b^g1_b）$发挥了重要作用。特别是，当$ g = gl_n（\ mathbb f_q）$是一个有限的通用线性组时，这种方法导致了$ q $ -schur代数$ a $ a $的北斗七星理论。这些代数可以通过$ \ sz：= \ mathbb z [t，t^{ - 1}] $作为内态代数的$ q $ -Analog（带有$ q = t^2 $），大于$ h $，涉及抛物线核心子群。代数$ a $是$ \ sz $上的准标准。类似的代数，仍然表示为$ a $，始终可以在其他类型中构建。但是，到目前为止，这些代数比$ gl_n $ case的用处不大，部分原因是它们通常不是准雌性。 几年前，作者对1998年的猜想进行了重新制定，提出（对于所有类型），存在具有分层的衍生模块类别的$ \ sz $ -Algebra $ a^+$，并通过kazhdan-lusztig细胞理论构建了阶层。代数$ a $作为$ a = ea^+e $恢复为a^+$中的dempotent $ e \。该专着的主要目的是完全证明这一猜想。该证明涉及使用精确类别的几种新的同源技术。遵循证明，我们表明$ a^+$在不良素数反转后确实变成了准雌性。提出了结果的一些首次应用 - 例如，用于分解矩阵 - 以及几个开放问题。
Let $G$ be a finite group of Lie type. In studying the cross-characteristic representation theory of $G$, the (specialized) Hecke algebra $H=\End_G(\ind_B^G1_B)$ has played a important role. In particular, when $G=GL_n(\mathbb F_q)$ is a finite general linear group, this approach led to the Dipper-James theory of $q$-Schur algebras $A$. These algebras can be constructed over $\sZ:=\mathbb Z[t,t^{-1}]$ as the $q$-analog (with $q=t^2$) of an endomorphism algebra larger than $H$, involving parabolic subgroups. The algebra $A$ is quasi-hereditary over $\sZ$. An analogous algebra, still denoted $A$, can always be constructed in other types. However, these algebras have so far been less useful than in the $GL_n$ case, in part because they are not generally quasi-hereditary. Several years ago, reformulating a 1998 conjecture, the authors proposed (for all types) the existence of a $\sZ$-algebra $A^+$ having a stratified derived module category, with strata constructed via Kazhdan-Lusztig cell theory. The algebra $A$ is recovered as $A=eA^+e$ for an idempotent $e\in A^+$. A main goal of this monograph is to prove this conjecture completely. The proof involves several new homological techniques using exact categories. Following the proof, we show that $A^+$ does become quasi-hereditary after the inversion of the bad primes. Some first applications of the result -- e.g., to decomposition matrices -- are presented, together with several open problems.
