Dynamical $R$ Matrices of Elliptic Quantum Groups and Connection Matrices for the $q$-KZ Equations

For any affine Lie algebra $\mathfrak g$, we show that any finite dimensional representation of the universal dynamical $R$ matrix $\mathcal R(\lambda)$ of the elliptic quantum group $\mathcal B_{q,\lambda}(\mathfrak g)$ coincides with a corresponding connection matrix for the solutions of the $q$-KZ equation associated with $U_q(\mathfrak g)$. This provides a general connection between $\mathcal B_{q,\lambda}(\mathfrak g)$ and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of $\mathcal R(\lambda)$ for $\mathfrak g=A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $D_n^{(1)}$, and show that they coincide with the face weights derived by Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.