  The goal of this paper is to lay the foundations for a combinatorial study,
via orthogonal functions and intertwining operators, of category O for the
rational Cherednik algebra of type G(r,p,n). As a first application, we give a
self-contained and elementary proof of the analog for the groups G(r,p,n), with
r>1, of Gordon's theorem (previously Haiman's conjecture) on the diagonal
coinvariant ring. We impose no restriction on p; the result for p<r has been
proved by Vale using a technique analogous to Gordon's. Because of the
combinatorial application to Haiman's conjecture, the paper is logically
self-contained except for standard facts about complex reflection groups. The
main results should be accessible to mathematicians working in algebraic
combinatorics who are unfamiliar with the impressive range of ideas used in
Gordon's proof of his theorem.
