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Intersection multiplicities and reflection subquotients of unitary reflection groups. I. (English) Zbl 0945.51005
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 181-193 (1999).
Let \(G\) be a finite complex \(n\)-dimensional reflection group. Let \(f_1,\ldots,f_n\) be basic homogeneous polynomial invariants for \(G\) of degree \(d_1,\ldots,d_n\), respectively. Fix a positive integer \(d\). Let \(E\) be an irreducible component of the variety \(\bigcap_{d\not d_i} f_i^{-1}(0)\). Then \(E\) is a linear subspace of \({\mathbb C}^n\) and every other irreducible component is conjugate to \(E\) under \(G\). The restriction to \(E\) of the stabilizer in \(G\) of \(E\) is a complex reflection group on \(E\), with basic homogeneous polynomial invariants \(f_i|_E\) for \(i\) such that \(d|d_i\). This is the main result of the paper.
Apart from the authors’ proof using intersection multiplicities, a proof by Looijenga using only basic algebraic geometry is given. The result extends parts of Springer’s earlier work on regular elements in [T. A. Springer, Invent. Math. 25, 159-198 (1974; Zbl 0287.20043)]. Some indications of applications, such as the decomposition of induced cuspidal representations of reductive groups over finite fields, are given.
For the entire collection see [Zbl 0910.00040].

51F15 Reflection groups, reflection geometries
20H15 Other geometric groups, including crystallographic groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
20G40 Linear algebraic groups over finite fields