arXiv 2105.005392021-05-02math.RT

Hopf Galois序上的Harish-Chandra模

Harish-Chandra modules over Hopf Galois orders

Jonas T. Hartwig

备注：: 22 pages, 1 figure

摘要：

伽罗瓦序理论是由Futorny和Ovsienko提出的。我们引入了$\mathcal{H}$ -Galois $Λ$ -序的概念.这些是在诺特整环$Λ$的分数域与Hopf代数$\mathcal{H}$（或者更一般地，与霍普夫代数的共理想子代数）的smash积中的某些非交换阶$F$。它们是韦伯斯特的主要旗帜秩序的概括。例子包括切雷德尼克代数，以及从霍普夫伽罗瓦论的例子。我们还定义了球面伽罗瓦序，它是作者引入的主伽罗瓦序的相应推广。主要结果是：（1）对于有限余维的$Λ$的每个极大理想$\mathfrak{m}$，在$\mathfrak{m}$的纤维中存在一个单Harish-Chandra $F$ -模;（2）对于$Λ$的每个特征标，我们构造了一个正则单Harish-Chandra模作为局部分布模的子商;（3）若某个稳定子余代数是有限维的，则对应的单Harish-Chandra模的纤维是有限的;（4）对称化幂等元的中心化子是球面伽罗瓦序，并且每个球面伽罗瓦序都是这样出现的。

英文摘要：

The theory of Galois orders was introduced by Futorny and Ovsienko. We introduce the notion of $\mathcal{H}$-Galois $Λ$-orders. These are certain noncommutative orders $F$ in a smash product of the fraction field of a noetherian integral domain $Λ$ by a Hopf algebra $\mathcal{H}$ (or, more generally, by a coideal subalgebra of a Hopf algebra). They are generalizations of Webster's principal flag orders. Examples include Cherednik algebras, as well as examples from Hopf Galois theory. We also define spherical Galois orders, which are the corresponding generalizations of principal Galois orders introduced by the author. The main results are (1) for every maximal ideal $\mathfrak{m}$ of $Λ$ of finite codimension, there exists a simple Harish-Chandra $F$-module in the fiber of $\mathfrak{m}$; (2) for every character of $Λ$ we construct a canonical simple Harish-Chandra module as a subquotient of the module of local distributions; (3) if a certain stabilizer coalgebra is finite-dimensional, then the corresponding fiber of simple Harish-Chandra modules is finite; (4) centralizers of symmetrizing idempotents are spherical Galois orders and every spherical Galois order appears that way.

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英文摘要：

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