Eigenvalues of quantum Gelfand invariants

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 ${\mathfrak{g}\mathfrak{l}}_n$. The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type $A$.