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A Proof of the $$q$$-Macdonald-Morris Conjecture for $$BC_n$$
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Memoirs of the American Mathematical Society
1994; 80 pp; softcover
Volume: 108
ISBN-10: 0-8218-2552-6
ISBN-13: 978-0-8218-2552-5
List Price: US$39 Individual Members: US$23.40
Institutional Members: US\$31.20
Order Code: MEMO/108/516

Macdonald and Morris gave a series of constant term $$q$$-conjectures associated with root systems. Selberg evaluated a multivariable beta type integral which plays an important role in the theory of constant term identities associated with root systems. Aomoto recently gave a simple and elegant proof of a generalization of Selberg's integral. Kadell extended this proof to treat Askey's conjectured $$q$$-Selberg integral, which was proved independently by Habsieger. This monograph uses a constant term formulation of Aomoto's argument to treat the $$q$$-Macdonald-Morris conjecture for the root system $$BC_n$$. The $$B_n$$, $$B_n^{\lor }$$, and $$D_n$$ cases of the conjecture follow from the theorem for $$BC_n$$. Some of the details for $$C_n$$ and $$C_n^{\lor }$$ are given. This illustrates the basic steps required to apply methods given here to the conjecture when the reduced irreducible root system $$R$$ does not have miniscule weight.

Research mathematicians.

• Introduction
• Outline of the proof and summary
• The simple roots and reflections of $$B_n$$ and $$C_n$$
• The $$q$$-engine of our $$q$$-machine
• Removing the denominators
• The $$q$$-transportation theory for $$BC_n$$
• Evaluation of the constant terms $$A,E,K,F$$ and $$Z$$
• $$q$$-analogues of some functional equations
• $$q$$-transportation theory revisited
• A proof of Theorem 4
• The parameter $$r$$
• The $$q$$-Macdonald-Morris conjecture for $$B_n,B_n^\lor ,C_n,C_n^\lor$$ and $$D_n$$
• Conclusion