Let $\ell$ be a prime and let $q$ be a prime power not divisible by $\ell$. Put $G=\mathrm{ GL}_n(\mathrm{ F}_q)$ and fix an irreducible cuspidal representation, $\bar{ \pi}$, of $G$ over a sufficiently large finite field, $k$, of characteristic $\ell$ 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 $\ell>n$. Our computations provide evidence for a conjecture of Helm relating the Bernstein center to the deformation theory of Galois representations.
Categories: math.NT