  We study components of the Bernstein category for a p-adic classical group
(with p odd) with inertial support a self-dual positive level supercuspidal
representation of a Siegel Levi subgroup. More precisely, we use the method of
covers to construct a Bushnell-Kutzko type for such a component. A detailed
knowledge of the Hecke algebra of the type should have number-theoretic
implications.
