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



Blocks of the category of representations of $GL_n({\mathbb F}_q)$
I may come a little late, but I think Theorem B' on page 52 of Bonnafé and Rouquiers' paper at IHES does what you want.


