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A proof of the q-Macdonald-Morris conjecture for BC n . (English) Zbl 0796.17003

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 ,,t n ) has a Laurent expansion at t 1 =0, then the constant term of f(t 1 ,,t n ) is fixed by t 1 qt 1 . The q-engine of the q-machine is the equivalent conclusion that q / 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.

Reviewer: A.Klimyk (Kiev)
MSC:
17B20Simple, semisimple, reductive Lie (super)algebras
33C80Connections of hypergeometric functions with groups and algebras