Let k be an algebraically closed field of characteristic $\ell$, and let $q = p^r$ be a prime power with $p \neq \ell$. Suppose I have a cuspidal representation $\pi$ of $GL_n({\mathbb F}_q)$, for some $n < \ell$.

The supercuspidal support of $\pi$ consists of $m$ copies of a supercuspidal representation $\sigma$ of $GL_{d}({\mathbb F}_q)$ for some integers $m$ and $d$ with $md = n$.

In this setting, there also exists a cuspidal representation $\pi'$ of $GL_m({\mathbb F}_{q^d})$ that has supercuspidal support equal to $m$ copies of the trivial character.

Let $A$ be the block of the category of $W(k)[GL_n({\mathbb F}_q)]$-modules containing $\pi$, and let $A'$ be the block of the category of $W(k)[GL_m({\mathbb F}_{q^d})]$-modules containing $\pi'$. Are $A$ and $A'$ equivalent as categories? (More precisely, is there an equivalence of $A$ and $A'$ that takes $\pi$ to $\pi'$?) I can prove this when $\pi$ is supercuspidal; i.e. $m=1$, but computations from character theory and considerations from the Langlands program make me suspect that it's true in general.

David Helm