On intersection indices of subvarieties in reductive groups

In this paper, I give an explicit formula for the intersection indices of the Chern classes (defined earlier by the author) of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups thus extending the beautiful result by D. Bernstein and Khovanskii, which holds for a complex torus. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classes of the compactification with hypersurfaces. The formula is similar to the Brion–Kazarnovskii formula for the intersection indices of hypersurfaces in reductive groups. The proof uses an algorithm of De Concini and Procesi for computing such intersection indices. In particular, it is shown that this algorithm produces the Brion–Kazarnovskii formula.