K-theorie equivariante des varietes de Bott-Samelson. Application a la structure multiplicative de la K-theorie equivariante des varietes de drapeaux

We construct a basis of the equivariant $K$-theory of Bott towers, and we describe precisely the multiplicative structure of these algebras. We deduce similar results for Bott-Samelson varieties. Thanks to the link between flag varieties and Bott-Samelson varieties, we give a method to compute the structure constants of the equivariant $K$-theory of flag varieties in the basis constructed by Kostant and Kumar.