Types in reductive p-adic groups: The Hecke algebra of a cover

In this paper, F is a non-Archimedean local field and G is the group of F-points of a connected reductive algebraic group defined over F. Also, tau is an irreducible representation of a compact open subgroup J of G, the pair (J; tau) being a type in G. The pair (J; tau) is assumed to be a cover of a type (J(L); tau (L)) in a Levi subgroup L of G. We give conditions, generalizing those of earlier work, under which the Hecke algebra H(G; tau) is the tensor product of a canonical image of H (L; tau (L)) and a sub-algebra H (K; tau), for a compact open subgroup K of G containing J.