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.