The projective envelope of a cuspidal representation of a finite linear group

Abstract Let l be a prime and let q be a prime power not divisible by l . Put G = GL n ( F q ) and fix an irreducible F ¯ l [ G ] -representation, π ¯ , such that π ¯ is cuspidal but not supercuspidal. We compute the W ( F ¯ l ) [ G ] -endomorphism ring of the projective envelope of π ¯ under the assumption that l > n > 1 . Our computations provide evidence for a conjecture of Helm relating the Bernstein center to the deformation theory of Galois representations (see [9] ).