A proof of the multi-component $q$-Baker--Forrester conjecture

The Selberg integral, an $n$-dimensional generalization of the Euler beta integral, plays a central role in random matrix theory, Calogero–Sutherland quantum many body systems, Knizhnik–Zamolodchikov equations, and multivariable orthogonal polynomial theory. The Selberg integral is known to be equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a $(p+1)$-component generalization of the $q$-Morris identity. It in turn yields a generalization of the Selberg integral. The $p=1$ case of Baker and Forrester’s conjecture was proved by Károlyi, Nagy, Petrov and Volkov in 2015. In this paper, we give a proof of the $(p+1)$-component $q$-Baker–Forrester conjecture, thereby settling this 26-year-old conjecture.