Equivariant K—theory of torus actions and formal characters

In this paragraph, we consider a torus T, that is a commutative connected reductive group over k, and a linear action of T on a vector space E of finite dimension r over k. We shall assume that all weights of T in E are positive with respect to some linear partial ordering ≥. We assume that the semi—group of positive (integral) weights in finitely generated. For example, T might be the group of homotheties of E. In the applications in subsequent chapters, T will be the maximal torus in a semisimple group, E will be the nilradical of a Borel subalgebra.