Homotopy type and -periodic homotopy groups of p-compact groups

We determine the v 1 -periodic homotopy groups of all irreducible p-compact groups ( B X , X ) . In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups.