Nested fibre bundles in Bott-Samelson varieties

We consider Bott-Samelson varieties BSc(s) for a semisimple compact Lie group C corresponding to sequences of (not necessarily simple) reflections s. Let n be the length s, K maximal torus in and W Weyl C. For any set R not overlapping integer pairs (i,j) such that 1?i?j?n function v:R?W, we subspace BSc(s,v)?BSc(s) solutions equations C/K requiring K-orbit product coordinates counted from i j equal v evaluated at (i,j)?R. decompose BSc(s,v) into twisted (in sense iterated fibre bundles) smaller BSc(t) fibres canonical projections flag variety. Finally, prove tensor decomposition K-equivariant cohomology BSc(s,v).