Semisimple strata for p-adic classical groups

Let F0 be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F0. In this paper, we prove some fundamental results towards the classification of the representations of G via the types of Bushnell and Kutzko. In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has previously been calculated.