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

Let ℓ be a prime and let q be a prime power not divisible by ℓ  . Put G=GLn(Fq)G=GLn(Fq) 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>1ℓ>n>1. Our computations provide evidence for a conjecture of Helm relating the Bernstein center to the deformation theory of Galois representations (see [9]).