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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].

##### MSC:
 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