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Zbl 0796.17003
Kadell, Kevin W.J.
A proof of the $q$-Macdonald-Morris conjecture for $BC\sb n$. (English)
[J] Mem. Am. Math. Soc. 516, 80 p. (1994). ISSN 0065-9266

{\it I. G. Macdonald} [SIAM J. Math. Anal. 13, 988-1007 (1982; Zbl 0498.17006)] and {\it 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 $BC\sb n$.\par The proof is based upon the fact that if $f(t\sb 1,\dots, t\sb n)$ has a Laurent expansion at $t\sb 1=0$, then the constant term of $f(t\sb 1,\dots, t\sb n)$ is fixed by $t\sb 1\to qt\sb 1$. The $q$-engine of the $q$-machine is the equivalent conclusion that $\partial\sb q/ \partial\sb q f$ has no residue at $t\sb 1=0$. The author uses an identity for a partial $q$-derivative which owes its existence to the geometry of the simple roots of $B\sb n$ and $C\sb n$. The author also requires certain antisymmetries of the terms occurring in the partial $q$-derivative and the $q$-transportation theory for $BC\sb n$. These are proved locally by using the basic properties of the simple reflections of $B\sb n$ and $C\sb n$. The author shows how to obtain the required functional equations using only the $q$-transportation theory for $BC\sb n$. This is based on the fact that $B\sb n$ and $C\sb n$ have the same Weyl group.
[A.Klimyk (Kiev)]

MSC 2000:: *17B20 Simple and semisimple Lie algebras
33C80 Connections of theory of special functions with groups and algebras

Keywords: Selberg beta integral; $q$-Macdonald-Morris conjecture; root system $BC\sb n$

Citations: Zbl 0498.17006

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