  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.
