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A Proof of the \(q\)-Macdonald-Morris Conjecture for \(BC_n\)
Kevin W. J. Kadell

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
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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.

Table of Contents

  • 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
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