Simple subquotients of big parabolically induced representations of p-adic groups

This note is motivated by the problem of “uniqueness of supercuspidal support” in the modular representation theory of p-adic groups. We show that any counterexample to the same property for a finite reductive group lifts to a counterexample for the corresponding unramified p-adic group. To this end, we need to prove the following natural property: any simple subquotient of a parabolically induced representation is isomorphic to a subquotient of the parabolic induction of some simple subquotient of the original representation. The point is that we put no finiteness assumption on the original representation.