Equivariant -theory of regular compactifications: further developments

Abstract

We describe the -equivariant -ring of , where is a factorial covering of a connected complex reductive algebraic group , and is a regular compactification of . Furthermore, using the description of , we describe the ordinary -ring as a free module (whose rank is equal to the cardinality of the Weyl group) over the -ring of a toric bundle over whose fibre is equal to the toric variety 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 and as algebras over and respectively, where is the wonderful compactification of the adjoint semisimple group . In the case when is a regular compactification of , we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over .