| Résume | This is a preliminary report on a work in progress. Coming from any formal group law, there is an oriented cohomology theory. It is expected that there is a quantum group associated to this formal group law, which acts on the corresponding cohomology theory applied to Nakajima quiver varieties. In this talk, I will describe some partial results in this direction, joint with Yang and Zhong. We prove that, for any formal group law, the formal affine Hecke algebra of Hoffnung-Malagon-Lopez-Savage-Zainoulline acts on the corresponding cohomology of the Springer fibers. Explicit formulae for the generalized Demazure-Lusztig operators are given. Also, we define a Hall algebra, generalizing the elliptic Hall algebra of Schiffmann-Vasserot, a.k.a., the shuffle algebra of Feigin-Odesskii. Then we prove that this algebra acts on the corresponding oriented cohomology of Nakajima quiver varieties. The Nakajima's raising operators are realized as elements in the Hall algebra. |
