Construction of eigenfunctions for a system of quantum minors of the monodromy matrix for an $SL(n,\mathbb C)$-invariant spin chain

We consider the problem of seeking the eigenvectors for a commuting family of quantum minors of the monodromy matrix for an $SL(n,\mathbb C)$-invariant inhomogeneous spin chain. The algebra generators and elements of the $L$-operator at each site of the chain are implemented as linear differential operators in the space of functions of $n(n{-}1)/2$ variables. In the general case, the representation of the $sl_n(\mathbb C)$ algebra at each site is infinite-dimensional and belongs to the principal unitary series. We solve this problem using a recursive procedure with respect to the rank $n$ of the algebra. We obtain explicit expressions for the eigenvalues and eigenvectors of the commuting family. We consider the particular cases $n=2$ and $n=3$ and also the limit case of the one-site chain in detail.