The projective envelope of a cuspidal representation of a GL[subscript n](F[subscript q])

Let l be a prime and let q be a prime power not divisible by l. Put G=GI[subscript n](F[subscript q])and fix a representation pi of G over a sufficiently large finite field, k, of characteristic l, so that pi is cuspidal but not supercuspidal. We compute the W(k)[G]-endomorphism ring of the projective envelope of pi under the assumption that l>n.