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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, τ is an irreducible representation of a compact open subgroup J of G, the pair (J, τ) being a type in G. The pair (J, τ) is assumed to be a cover of a type (JL, τL) in a Levi subgroup L of G. We give conditions, generalizing those of earlier work, under which the Hecke algebra H(G, τ) is the tensor product of a canonical image of H(L, τL) and a sub-algebra H(K, τ), for a compact open subgroup K of G containing J . We return to our earlier paper [2] in order to simplify and generalize one of its more suggestive results. 1.1. Let F be a non-Archimedean local field and G the group of F -rational points of a connected reductive algebraic group defined over F . We write R(G) for the category of all smooth representations of G. As in [2], §1, we form the set B(G): this consists of G-inertial equivalence classes of pairs (L, σ), where L is an F -Levi subgroup of G and σ is an irreducible supercuspidal representation of L. As in [2], 1.4, and [1], 2.10, this leads to a canonical decomposition

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