The Projective Envelope of a Cuspidal Representation of a Finite Linear Group

Let $\mathrm{\ell }$ be a prime and let $q$ be a prime power not divisible by $\mathrm{\ell }$. Put $G={\mathrm{G}\mathrm{L}}_n({\mathrm{F}}_q)$ and fix an irreducible cuspidal representation, $\bar{\pi }$, of $G$ over a sufficiently large finite field, $k$, of characteristic $\mathrm{\ell }$ such that $\bar{\pi }$ is not supercuspidal. We compute the $\mathrm{W}(k)[G]$-endomorphism ring of the projective envelope of $\bar{\pi }$ under the assumption that $\mathrm{\ell }>n$. Our computations provide evidence for a conjecture of Helm relating the Bernstein center to the deformation theory of Galois representations.