Towards a Schubert calculus for complex reflection groups

Abstract

The cohomology ring of the flag manifold associated to any compact Lie group has a simple description in terms of the Weyl group of $G$. The same algebraic procedure gives a ring associated to any complex re ection group, not necessarily a Weyl group, but much less is known about it. For example, in the case of Weyl groups (and the corresponding compact Lie groups), there are remarkably simple formulas for the degrees of Grassmannians and related projective varieties, due to Schubert and others. We find here a similar formula for the ‘degree’ of a ring associated to the complex re ection group $S_n**{\bf Z}/e$; in the case $e = 2$, this group is the Weyl group of the symplectic group and our formula specializes to the classical formula for the degree of the isotropic Grassmannian $Sp(2n)/U(n)$. We discuss the relation of our calculation to Hall–Littlewood polynomials at roots of unity.