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

Abstract

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