Nested fibre bundles in Bott-Samelson varieties.

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