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 K\'{a}rolyi, 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.