Rational tori, semisimple orbits and the topology of hyperplane complements
Let \(G\) be a connected reductive group defined over the finite field \(\mathbb{F}_ q\) of \(q\) elements. Let \(F\) be the associated Frobenius endomorphism of \(G\). The author shows how to count regular semisimple classes in \(G^ F\) (resp. orbits in the Lie algebra \({\mathfrak g}^ F\)). If \(G\) is semisimple, he shows that the number of regular semisimple classes of \(G^ F\) is always odd, regardless of the characteristic, and the number of unipotent (resp. nilpotent) elements in \(G^ F\) (resp. \({\mathfrak g}^ F\)) is equal to the number of \(F\)-stable maximal tori of \(G^ F\), which is equal to \(q^{2N}\), where \(N\) is the number of positive roots of \(G\). Moreover, he shows that if \(\varepsilon\) is the alternating character of \(W\), the Weyl group of \(G\), and \(M_ W\) is the complexified hyperplane complement corresponding to \(W\), then \(\varepsilon\) does not occur in the cohomology modules \(H^*(M_ W)\), which generalises Stanley's result.
scientific article; zbMATH DE number 93806