Construction of the Gelfand–Tsetlin basis for unitary principal series representations of the algebra $sl_n(\mathbb C)$

We consider infinite-dimensional unitary principal series representations of the algebra $sl_n(\mathbb C)$, implemented on the space of functions of $n(n{-}1)/2$ complex variables. For such representations, the elements of the Gelfand–Tsetlin basis are defined as the eigenfunctions of a certain system of quantum minors. The parameters of these functions, in contrast to the finite-dimensional case, take a continuous series of values. We obtain explicit formulas that allow constructing these functions recursively in the rank of the algebra $n$. The main construction elements are operators intertwining equivalent representations and also a group operator of a special type. We demonstrate how the recurrence relations work in the case of small ranks.