For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ of $B$, and the differential operators on $E$ are closely linked to the representation theory of $G$.

For the special case of a flag manifold, which is to say, when $B$ is a Borel subgroup of $G$, differential operators from $E$ to itself correspond to homomorphisms of the Verma module $U({\frak g})\otimes_{U({\frak b})} V_{\lambda}$. These homomorphisms are in turn classified by the so-called **singular vectors** of $V_{\lambda}$, which is to say the vectors killed by the action of the positive niradical. Moreover again, these singular vectors correspond to solutions of certain hyper-geometric functions.

What I would like to know is how all this relates to equivariant K-theory. Is there some characterization of the singular vectors correspond to a Fredholm operator. Also, can the defining equivalence relation of the equivariant K-theory group $K^0$ be nicely reformulated in terms of representation theory and singular vectors?