Rational points and cohomology of discriminant varieties
If \(G\) is a unitary reflection group acting on an \(n\)-dimensional complex vector space \(V\), we let \(X\) denote the space of regular orbits of \(G\); that is, we remove the reflecting hyperplanes from \(V\), and consider the orbit space of \(G\) on what remains. \(X\) is called the discriminant variety of \(G\); when \(G\) is the symmetric group on the \(n\) coordinates, \(X\) is well-known to be isomorphic to the space of complex polynomials in one variable with distinct roots. Let \(\tilde{G}\) be a subgroup of GL\((V)\) which normalises \(G\). Then \(\tilde{G}\) acts on \(X\), and on its cohomology. In the paper under review, the author computes the cohomology action. One can easily restrict oneself to the case where \(G\) is irreducible, and there is a classification due to \textit{G. Shephard} and \textit{J. Todd} [Can. J. Math. 6, 274--304 (1954; Zbl 0055.14305)] of such groups. There is one generic class and six exceptional cases. For the generic cases, the computation amounts to the computation of the cohomology of classical discriminant varieties, with local coefficient systems. The results are given explicitly as formulae for the equivariant Poincaré polynomial, inside the polynomial ring in one variable over the Grothendieck ring of representations of \(\tilde{G}/G\). Some of the computations of cohomology involve using a comparison between the cohomology groups of interest and \(\ell\)-adic cohomology groups, and then using the Grothendieck fixed-point formula to relate this to the number of points on (the reduction of) \(X\) over finite fields. This approach continues that of \textit{M. Kisin} and the author [J. Algebra 247, 435--451 (2002; Zbl 1039.14005)].
scientific article; zbMATH DE number 2105772