THE COHOMOLOGY OF THE REGULAR SEMISIMPLE VARIETY

We use the equivariant cohomology of hyperplane complements and their toral counterparts to give formulae for the Poincare polynomials of th e varieties of regular semisimple elements of a reductive complex Lie group or Lie algebra. As a result, we obtain vanishing theorems for ce rtain of the Betti numbers. Similar methods, using l-adic cohomology, may be used to compute numbers of rational points of the varieties ove r the finite field F-q. In the classical cases, one obtains, both for the Poincare polynomials and for the numbers of rational points, polyn omials which exhibit certain regularity conditions as the dimension in creases. This regularity may be interpreted in terms of functional equ ations satisfied by certain power series, or in terms of the represent ation theory of the Weyl group. (C) 1998 Academic Press.