  We consider the quantum Gelfand invariants which first appeared in a landmark
paper by Reshetikhin, Takhtadzhyan and Faddeev (1989). We calculate the
eigenvalues of the invariants acting in irreducible highest weight
representations of the quantized enveloping algebra for ${\mathfrak {gl}}_n$.
The calculation is based on Liouville-type formulas relating two families of
central elements in the quantum affine algebras of type $A$.
