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.