Eigenvalues of quantum Gelfand invariants

Abstract

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 $A$.

13 pages, construction of central elements and some proofs were simplified in v2, limit values as q->1 are discussed in v3, reference to earlier work on Liouville formula added in v4