  We introduce a category $\mathcal O$ of representations of the elliptic
quantum group associated with $\mathfrak{sl}_2$ with well-behaved $q$-character
theory. We derive separation of variables relations for asymptotic
representations in the Grothendieck ring of this category. Baxter $Q$-operators
are obtained as transfer matrices for asymptotic representations and obey
$TQ$-relations as a consequence of the relations in $K_0(\mathcal O)$.
