The Cohomology of the Regular Semisimple Variety Original Research Article

We use the equivariant cohomology of hyperplane complements and their toral counterparts to give formulae for the Poincaré polynomials of the varieties of regular semisimple elements of a reductive complex Lie group or Lie algebra. As a result, we obtain vanishing theorems for certain of the Betti numbers. Similar methods, usingl-adic cohomology, may be used to compute numbers of rational points of the varieties over the finite field imageq. In the classical cases, one obtains, both for the Poincaré polynomials and for the numbers of rational points, polynomials which exhibit certain regularity conditions as the dimension increases. This regularity may be interpreted in terms of functional equations satisfied by certain power series, or in terms of the representation theory of the Weyl group.