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

TL;DR

This paper computes the endomorphism ring of the projective envelope of a specific cuspidal representation of a finite general linear group, providing evidence for a conjecture linking Bernstein centers and Galois deformation theory.

Contribution

It explicitly determines the endomorphism ring of the projective envelope of a non-supercuspidal cuspidal representation, supporting a conjecture connecting Bernstein centers to Galois deformation theory.

Findings

Computed the endomorphism ring of the projective envelope for certain cuspidal representations.

Provided evidence supporting Helm's conjecture relating Bernstein center and Galois deformation theory.

Established results under the condition that  > n.

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.