The elliptic quantum group and geometric representation conjectures

Let $U_{q,p}(\widehat{\mathfrak g}])$ and $U_{t_1,t_2,p}(\mathfrak g_{tor})$ be elliptic quantum groups associated with a Dynkin type $\mathfrak g$, and let $X$ be the corresponding Dynkin quiver variety in the first case or affine Dynkin quiver variety in the second. Let $\mathrm{E}_\mathrm{T}(X)$ denote its torus-equivariant elliptic cohomology, and let elliptic stable envelopes be the integration kernels of screened vertex operators. The highest-to-highest expectation value is the matrix coefficient between highest-weight vectors, and a quasimap is a map $\mathbb P^1\dashrightarrow X$.

Elliptic quantum group and geometric representation conjectures. (1) Vertex operators of $U_{q,p}(\widehat{\mathfrak g})$ or $U_{t_1,t_2,p}(\mathfrak g_{tor})$ defined as intertwining operators with respect to the standard comultiplication are realized as screened vertex operators with the elliptic stable envelope for $\mathrm{E}_\mathrm{T}(X)$ as their integration kernel; in particular, their components are labeled by the $\mathrm{T}$-fixed points of $X$. (2) Compositions of these vertex operators give the shuffle product formula for elliptic stable envelopes as a relation among the integration kernels. (3) The highest-to-highest expectation values of compositions of vertex operators give the $\mathrm{K}$-theoretic vertex functions counting quasimaps $\mathbb P^1\dashrightarrow X$.

These conjectures propose a compatibility between elliptic quantum-group representations and the geometry of quiver varieties, including stable envelopes, shuffle products, and quasimap-counting vertex functions. The source does not state a resolution of these claims.