  We introduce a category O of modules over the elliptic quantum group of sl_N
with well-behaved q-character theory. We construct asymptotic modules as
analytic continuation of a family of finite-dimensional modules, the
Kirillov--Reshetikhin modules. In the Grothendieck ring of this category we
prove two types of identities: generalized Baxter relations in the spirit of
Frenkel--Hernandez between finite-dimensional modules and asymptotic modules;
three-term Baxter TQ relations of infinite-dimensional modules.
