  We solve the problem of extension of characters of commutative subalgebras in
associative (noncommutative) algebras for a class of subrings (Galois orders)
in skew group rings. These results can be viewed as a noncommutative analogue
of liftings of prime ideals in the case of integral extensions of commutative
rings. The proposed approach can be applied to the representation theory of
many infinite dimensional algebras including universal enveloping algebras of
reductive Lie algebras, Yangians and finite $W$-algebras. In particular, we
develop a theory of Gelfand-Tsetlin modules for $\gl_n$. Besides classification
results we characterize their categories in the generic case extending the
classical results on $\gl_2$.
