Persitence of non degeneracy: a local analog of Iharas lemma

Persitence of non degeneracy is a phenomenon which appears in the theory of $overline{mathbb Q}_l$-representations of the linear group: every irreducible submodule of the restriction to the mirabolic subgroup of an non degenerate irreducible representation is non degenerate. This is no more true in general, if we look at the modulo $l$ reduction of some stable lattice. As in the Clozel-Harris-Taylor generalization of global Iharas lemma, we show that this property, called non degeneracy persitence, remains true for lattices given by the cohomology of Lubin-Tate spaces.