Equivariant K-theory of regular compactifications: further developments

摘要

We describe the (G) over tilde x (G) over tilde -equivariant K-ring of X, where (G) over tilde is a factorial covering of a connected complex reductive algebraic group G, and X is a regular compactification of G. Furthermore, using the description of K-(G) over tildex (G) over tilde(X), we describe the ordinary K-ring K(X) as a free module (whose rank is equal to the cardinality of the Weyl group) over the K-ring of a toric bundle over G/B whose fibre is equal to the toric variety (T) over bar (+) associated with a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [1]). We also give an explicit presentation of K-(G) over tildex (G) over tilde(X) and K(X) as algebras over K-(G) over tildex (G) over tilde((G(ad)) over bar) and K((G(ad)) over bar) respectively, where (G(ad)) over bar is the wonderful compactification of the adjoint semisimple group G(ad). In the case when X is a regular compactification of G(ad), we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over (G(ad)) over bar.