The Selberg–Jack Symmetric Functions

Abstract

K. Aomoto has recently given a simple proof of an extension of A. Selberg's integral. We prove the following generalization of Aomoto's theorem. For each k⩾0, there exists a family { s k λ ( t)} of homogeneous symmetric functions such that if the integrand of Selberg's integral is multiplied by s k λ ( t 1, …, t n ), then the integral has a certain closed form. Since s k λ ( t) is a renormalized Jack symmetric function with α=1/ k, we call { s k λ ( t)} the Selberg–Jack symmetric functions. R. P. Stanley and I. G. Macdonald have extended many of the basic properties of the I. J. Schur functions to the Jack symmetric functions. Stanley's extensions of the Pieri formula, in a form obtained by Macdonald's extension of the duality property, and the combinatorial representation arise naturally in our proof. We give proofs of these and other results of Stanley and Macdonald. We give two constant-term identities which are equivalent to the integration formula. These results extend W. G. Morris' theorem for A n and the Macdonald–Morris constant term identity for BC n . By exploring Selberg's proof of his integral, we give a recurrence relation, outline a method for calculating the Selberg–Jack symmetric functions, and show that a constant term orthogonality relation, recently proved by Macdonald, arises naturally. We give an expansion which allows us to evaluate the normalization of this orthogonality, thus giving an alternative proof of the integration formula.