The Gelfand-Tsetlin basis for infinite-dimensional representations of $gl_n(\mathbb{C})$

We consider the problem of determination of the Gelfand-Tsetlin basis for unitary principal series representations of the Lie algebra $gl_n(\mathbb{C})$. The Gelfand-Tsetlin basis for an infinite-dimensional representation can be defined as the basis of common eigenfunctions of corner quantum minors of the corresponding L-operator. The construction is based on the induction with respect to the rank of the algebra: an element of the basis for $gl_n(\mathbb{C})$ is expressed in terms of a Mellin-Barnes type integral of an element of the basis for $gl_{n-1}(\mathbb{C})$. The integration variables are the parameters (in other words, the quantum numbers) setting the eigenfunction. Explicit results are obtained for ranks $3$ and $4$, and the orthogonality of constructed sets of basis elements is demonstrated. For $gl_3(\mathbb{C})$ the kernel of the integral is expressed in terms of gamma-functions of the parameters of eigenfunctions, and in the case of $gl_4(\mathbb{C})$ -- in terms of a hypergeometric function of the complex field at unity. The formulas presented for an arbitrary rank make it possible to obtain the system of finite-difference equations for the kernel. They include expressions for the quantum minors of $gl_n(\mathbb{C})$ L-operator via the minors of $gl_{n-1}(\mathbb{C})$ L-operator for the principal series representations, as well as formulas for action of some non-corner minors on the eigenfunctions of corner ones. The latter hold for any representation of $gl_n(\mathbb{C})$ (not only principal series) in which the corner minors of the L-operator can be diagonalized.