The Gelfand–Tsetlin basis for infinite-dimensional representations of gln(C)

Abstract

Abstract We consider the problem of determination of the Gelfand–Tsetlin basis for unitary principal series representations of the Lie algebra g l n ( 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 g l n ( C ) is expressed in terms of a Mellin–Barnes type integral of an element of the basis for g l n − 1 ( 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 g l 3 ( C ) the kernel of the integral is expressed in terms of gamma-functions of the parameters of eigenfunctions, and in the case of g l 4 ( 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 g l n ( C ) L-operator via the minors of g l n − 1 ( 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 g l n ( C ) (not only principal series) in which the corner minors of the L-operator can be diagonalized.