# zbMATH — the first resource for mathematics

Twisted invariant theory for reflection groups. (English) Zbl 1172.20030
Let $$G$$ be a finite reflection group acting in a finite dimensional complex vector space $$V$$ and let $$S$$ be the coordinate ring of $$V$$. Any element $$\gamma\in\text{GL}(V)$$ which normalizes $$G$$ acts on the ring $$S^G$$ of $$G$$-invariants. For any finite dimensional $$\langle G,\gamma\rangle$$-module $$M$$ the authors associate with the coset $$G\gamma$$ a set of $$m=\dim M$$ constants $$\varepsilon_i(M)$$. Various algebraic and geometric properties of $$G\gamma$$ may be expressed in terms of the constants $$\varepsilon_i$$. If $$G'$$ is a parabolic subgroup of $$G$$ which is normalized by $$\gamma$$, then the constants $$\varepsilon_i(M)$$ are the same whether $$M$$ is regarded as a module for $$\langle G,\gamma\rangle$$ or $$\langle G',\gamma\rangle$$. This idea is behind many of the results and their proofs.
First applications are to the notion of regular elements and regular eigenvalues for the reflection cosets. A vector $$v\in V$$ is ($$G$$-)regular if $$v$$ does not lie on any reflecting hyperplane of $$G$$. The element $$\gamma$$ of $$G\gamma$$ is regular if it has a regular eigenvector $$v$$; if $$\gamma v=\zeta v$$, then $$\zeta$$ is called a regular eigenvalue. The authors give precise criteria for an eigenvalue to be regular for a coset and show that, under obvious necessary qualifications, the coset $$G\gamma$$ contains a regular element.
As a further application, a generalization of a result of Springer-Stembridge which relates the module structure of the coinvariant algebras of $$G$$ and $$G'$$ is obtained. Also the existence of analogues of Coxeter elements of the real reflection groups in the so-called well-generated groups is proved and quotients of $$G$$ which are themselves reflection groups are studied.

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 13A50 Actions of groups on commutative rings; invariant theory 51F15 Reflection groups, reflection geometries 20H15 Other geometric groups, including crystallographic groups
Full Text: