ABSTRACT:

It was shown by M.Aganagic and A.Okounkov that the relative insertions in the quantum K-theory of Nakajima varieties are equivalent to non-singular descendent insertions. Among other things, this result leads to integral representation for solutions of qKZ equations and explicit description of the Bethe vectors.
      
From representation theoretic viewpoint, the result of Aganagic-Okoukov deals with the “slope” zero qKZ. In my talk I discuss generalizations of qKZ equations to an arbitrary slope. I also discuss the result of H. Dinkins describing the integral solutions of qKZ equations and the corresponding Bethe vectors for general slopes.  These results point to existence of an enumerative theory generalizing the relative quasimap count for Nakajima varieties.