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 -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 - Macdonald-Morris conjecture for the root system .
The proof is based upon the fact that if has a Laurent expansion at , then the constant term of is fixed by . The -engine of the -machine is the equivalent conclusion that has no residue at . The author uses an identity for a partial -derivative which owes its existence to the geometry of the simple roots of and . The author also requires certain antisymmetries of the terms occurring in the partial -derivative and the -transportation theory for . These are proved locally by using the basic properties of the simple reflections of and . The author shows how to obtain the required functional equations using only the -transportation theory for . This is based on the fact that and have the same Weyl group.