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We construct a finite dimensional representation of the face type, i.edynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlinbasis of the tensor product of the $n$-vector representations. The result isdescribed in a combinatorial way by using the partitions of $[1,n]$. We findthat the change of basis matrix from the standard to the Gelfand-Tsetlin basisis given by a specialization of the elliptic weight function obtained in theprevious paper[Konno17]. Identifying the elliptic weight functions with theelliptic stable envelopes obtained by Aganagic and Okounkov, we show acorrespondence of the Gelfand-Tsetlin bases (resp. the standard bases) to thefixed point classes (resp. the stable classes) in the equivariant ellipticcohomology $E_T(X)$ of the cotangent bundle $X$ of the partial flag variety. Asa result we obtain a geometric representation of the elliptic quantum group on$E_T(X)$.

Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group