Equivariant \(K\)-theory of flag Bott manifolds of general Lie type (Q6594783)

This article computes the \(K\)-theory and the equivariant \(K\)-theory of certain spaces with torus actions, called generalized Bott manifolds. These are defined recursively by taking fibre bundles in each step, where the fibres are homogeneous spaces such as flag manifolds. The final results compute the \(K\)-theory and the equivariant \(K\)-theory as rings very explicitly, using representation rings of the relevant Lie groups as the main ingredient. These computations become even more explicit when the relevant Lie groups are \(\mathrm{Sl}(n, \mathbb{C})\), where they are expressed using rings of invariant polynomials. The article ends with some concrete examples involving other complex semisimple Lie groups.\N\NThe most basic results in the article concern a variety with an action of the complex torus \((\mathbb{C}^*)^n\), equipped with a filtration where the difference sets are affine spaces with a linear representation of the torus. More generally, certain principal bundles of such varieties are considered. Under the assumptions in the article, the \(K\)-theory may be computed explicitly for such spaces. The key insight is that certain long exact sequences split into many short exact sequences.