I. G. Macdonald [SIAM J. Math. Anal. 13, 988-1007 (1982; Zbl 0498.17006)] and W. G. Morris [Ph. D. dissertation, Univ. Wisconsin, Madison (1982)] gave a series of constant term \(q\)-conjectures associated with root systems. Selberg evaluated a multivariate beta integral which plays an important role in the theory of constant term identities associated with root systems. Aomoto gave a simple proof of a generalization of the Selberg integral. The author of the present paper uses a constant term formulation of Aomoto’s argument to treat the \(q\)- Macdonald-Morris conjecture for the root system \(BC_ n\).
The proof is based upon the fact that if \(f(t_ 1,\dots, t_ n)\) has a Laurent expansion at \(t_ 1=0\), then the constant term of \(f(t_ 1,\dots, t_ n)\) is fixed by \(t_ 1\to qt_ 1\). The \(q\)-engine of the \(q\)-machine is the equivalent conclusion that \(\partial_ q/ \partial_ q f\) has no residue at \(t_ 1=0\). The author uses an identity for a partial \(q\)-derivative which owes its existence to the geometry of the simple roots of \(B_ n\) and \(C_ n\). The author also requires certain antisymmetries of the terms occurring in the partial \(q\)-derivative and the \(q\)-transportation theory for \(BC_ n\). These are proved locally by using the basic properties of the simple reflections of \(B_ n\) and \(C_ n\). The author shows how to obtain the required functional equations using only the \(q\)-transportation theory for \(BC_ n\). This is based on the fact that \(B_ n\) and \(C_ n\) have the same Weyl group.