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

Abstract

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.