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?

  • $\begingroup$ 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? $\endgroup$ – Vít Tuček May 22 '13 at 22:00
  • $\begingroup$ 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. $\endgroup$ – Jean Delinez May 23 '13 at 10:48
  • $\begingroup$ 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. $\endgroup$ – Vít Tuček May 23 '13 at 11:04
  • $\begingroup$ Yes, the is true. Some good references can be found here mathoverflow.net/questions/109392/… $\endgroup$ – Jean Delinez May 23 '13 at 11:40
  • $\begingroup$ I meant the stuff about hyper-geometric functions. Sorry, I should have been more specific. $\endgroup$ – Vít Tuček May 23 '13 at 19:25

The ideas were first developed by Givental and Lee in the context of quantum equivariant K-theory https://arxiv.org/abs/math/0108105, where they defined quantum K-theory as a certain lift of quantum cohomology. Then their results were generalized from flag varieties to Nakajima quiver varieties (at least of type A).

So to answer the first part of your question the (difference or q-) hypergemetric functions are (up to a certain normalization) singular vectors that you have mentioned. Based on what we know from Givental, Lee and the follow-up fork is that those functions are nothing but K-theoretic J-functions of the corresponding variety. One can also check explicitly that they reproduce quantum K-ring relations in the semiclassical limit.

I am gonna give an example for $T^*\mathbb{P}^1$ (which you can find in 3.5.3 of https://arxiv.org/pdf/1412.6081.pdf). So the corresponding singular vector is a hypergeometric function of type $_2\phi_1$ and it satisfies a difference equation (which is an integral of motion of some integrable system). The normalization factor consisting of the ratio of theta functions (formula (3.42)) has a meaning of its own and is called $\textit{elliptic envelope}$ by Aganagic and Okounkov https://arxiv.org/abs/1604.00423.


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