Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

We construct a finite dimensional representation of the face type, i.e. dynamical, elliptic quantum group associated with |slN| on the Gelfand-Tsetlin basis of the tensor product of the |n|-vector representations. The result is described in a combinatorial way by using the partitions of |$[1,n]$|⁠. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (respectively the standard bases) to the fixed point classes (respectively the stable classes) in the equivariant elliptic cohomology |ET(X)| of the cotangent bundle |X| of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on |ET(X)|⁠. Communicated by: Prof.