Let $G$ be a finite group generated by (pseudo-) reflections in acomplex vector space and let $g$ be any linear transformation whichnormalises $G$. In an earlier paper, the authors showed how toassociate with any maximal eigenspace of an element of the coset$gG$, a subquotient of $G$ which acts as a reflection group on theeigenspace. In this work, we address the questions ofirreducibility and the coexponents of this subquotient, as well ascentralisers in $G$ of certain elements of the coset. A criterionis also given in terms of the invariant degrees of $G$ for aninteger to be regular for $G$. A key tool is the investigation ofextensions of invariant vector fields on the eigenspace, whichleads to some results and questions concerning the geometry ofintersections of invariant hypersurfaces.
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机译：令$ G $为复矢量空间中（伪）反射生成的有限群，令$ g $为使$ G $归一化的任何线性变换。在较早的论文中，作者展示了如何与coset $ gG $的元素的任何最大特征空间相关联，该元素是$ G $的子商，它充当特征空间上的反射组。在这项工作中，我们解决了不可约性和该次商的共指数的问题，以及同伴集某些元素的$ G $的集中化问题。还针对$ G $的正整数，给出了$ G $不变度的标准。关键工具是研究本征空间上不变矢量场的扩展，这导致了一些关于不变超曲面相交的几何形状的结果和问题。
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