The $\bal$\ and $\bcl$\ Bailey Transform and Lemma

Abstract

We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative ``Bailey chain'' concept in the setting of basic hypergeometric series very well-poised on unitary $A_{\ell}$ or symplectic $C_{\ell}$ groups. The classical case, corresponding to $A_1$ or equivalently $\roman U(2)$, contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. In particular, our program extends much of the classical work of Rogers, Bailey, Slater, Andrews, and Bressoud.