Twisted invariant theory for reflection groups.

*(English)*Zbl 1172.20030Let \(G\) 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.

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.

Reviewer: Ivan V. Arzhantsev (Moskva)