Cycles of Bott-Samelson type for taut representations

Abstract

Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut, is either variationally complete or it is one of the following orthogonal representations (n bigger than or equal to 2): the (standard) x_R (spin) representation of SO(2)xSpin(9); or the (standard) x_C (standard) representation of U(2)xSp(n); or the (standard)^3 x_ H (standard) representation of SU(2)xSp(n). In this paper we will show how to adapt the construction of the cycles of Bott and Samelson to the orbits of these three representations. As a result, they also admit explicit cycles representing a basis of their Z_2-homology and, in particular, this provides another proof of their tautness.