Kadell’s two conjectures for Macdonald polynomials

Abstract

Recently Kevin Kadell found interesting properties of anti-symmetric variants of the so-called Jack polynomials [Ka].He formulated two conjectures about negative integral and half-integral values of the parameter k (k = 1 for the characters of compact simple Lie groups).As it was observed independently by Ian Macdonald and the author, these conjectures follow readily from the interpretation of the Jack polynomials as eigenfunctions of the Calogero-Sutherland -Heckman -Opdam second order operators generalizing the radial parts of the Laplace operators on symmetric spaces (see [HO,He,M1,M2]).The difference case requires a bit different treatment but still is not complicated.We will formulate and prove the Kadell conjectures for the Macdonald polynomials (the q, t-case).These statements are of certain interest because negative k are somehow connected with irreducible represenations for anti-dominant highest weights (and with represenations of Kac-Moody algebras of negative integral central charge).They also make more complete the theory of Macdonlad's polynomials at roots of unity started in [Ki], [C3,C4].Half-integral k appear in the theory of spherical functions.Anyway it is challenging to understand what is going on when 0 > k ∈ Q, since these values are singular for the coefficients of symmetric Macdonald polynomials.