We introduce and study a category $text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_qmathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional $U_qmathfrak{g}$ modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in $text{Fin}$. Among them we find the Baxter $Q_i$ operators and $T_i$ operators satisfying relations of the form $T_iQ_i=prod_j Q_j+ prod_k Q_k$. We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the $Q_i$ operators acting in an arbitrary finite-dimensional representation of $U_qmathfrak{g}$.
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