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A proof of the $q$-Macdonald-Morris conjecture for $B{C}_{n}$. (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 $B{C}_{n}$.

The proof is based upon the fact that if $f\left({t}_{1},\cdots ,{t}_{n}\right)$ has a Laurent expansion at ${t}_{1}=0$, then the constant term of $f\left({t}_{1},\cdots ,{t}_{n}\right)$ is fixed by ${t}_{1}\to q{t}_{1}$. 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 $B{C}_{n}$. 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 $B{C}_{n}$. This is based on the fact that ${B}_{n}$ and ${C}_{n}$ have the same Weyl group.

Reviewer: A.Klimyk (Kiev)
##### MSC:
 17B20 Simple, semisimple, reductive Lie (super)algebras 33C80 Connections of hypergeometric functions with groups and algebras