# Equivariant $K$-theory, singular vectors, and flag manifolds

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?

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You confused notation in the first paragraph. The bundle should be over $M$ and $H$ should be $B$. Is there any reason why don't you suppose that $G/B$ is a flag variety right from the beginning? –  Vít Tuček May 22 '13 at 22:00
Sorry, that's been fixed now. And no there is no reason why I don't assume $G/B is a flag variety from the start. – Jean Delinez May 23 '13 at 10:48 OK. I just wonder whether the second paragraph can be pushed to generalized flag manifolds$G/P$for$P\$ parabolic. Do you know any references here? At least for the Borel case. –  Vít Tuček May 23 '13 at 11:04
Yes, the is true. Some good references can be found here mathoverflow.net/questions/109392/… –  Jean Delinez May 23 '13 at 11:40
I meant the stuff about hyper-geometric functions. Sorry, I should have been more specific. –  Vít Tuček May 23 '13 at 19:25