# A Proof of the \(q\)-Macdonald-Morris Conjecture for \(BC_{n}\)

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*Kevin W. J. Kadell*

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.

#### Table of Contents

# Table of Contents

## A Proof of the $q$-Macdonald-Morris Conjecture for $BC_{n}$

- Table of Contents v6 free
- 1. Introduction 18 free
- 2. Outline of the proof and summary 613 free
- 3. The simple roots and reflections of B[sub(n)] and C[sub(n)] 1623
- 4. The g-engine of our q-machine 2532
- 5. Removing the denominators 3239
- 6. The q-transportation theory for BC[sub(n)] 3643
- 7. Evaluation of the constant terms A, E, K, F and Z 4249
- 8. q-analogues of some functional equations 4956
- 9. g-transportation theory revisited 5663
- 10. A proof of Theorem 4 6067
- 11. The parameter r 6976
- 12. The g-Macdonald-Morris conjecture for B[sub(n)], B[sup(v)][sub(n)], C[sub(n)], C[sup(v)][sub(n)] and D[sub(n)] 7178
- 13. Conclusion 7885