Finite-Dimensional Orthogonality Structures for Hall-Littlewood Polynomials

We present a finite-dimensional system of discrete orthogonality relations for the Hall-Littlewood polynomials. A compact determinantal formula for the weights of the discrete orthogonality measure is formulated in terms of a Gaudin-type conjecture for the normalization constants of a dual system of orthogonality relations. The correctness of our normalization conjecture has been checked in some special cases: for Hall-Littlewood polynomials up to four variables (i), for the reduction to Schur polynomials (ii), and in a continuum limit in which the Hall-Littlewood polynomials degenerate into the Bethe Ansatz eigenfunctions of the Schrödinger operator for identical Bose particles on the circle with pairwise delta-potential interactions (iii).