Let $H_n(q,k)$ be the Iwahori Hecke algebra of symmetric group $S_n$ over an algebraically closed field $k$ of characteristic $p>0$, where $q$ is an invertable element in $k$. Assume that $q$ is a primitive $e$-th root of unity.
In the case $q$=power of some prime $l$ (mod p) with $l\neq p$, there is an isomorphism between $H_n(q,k)$ and $\operatorname{End}_{kG}(1_B^G)$, where $G$ is finite linear group over finite field $\mathbb{F}_q$，$B$ is Borel subgroup of $G$, $1_B^G$ is induction of trivial $B$-module to $G$.
My question is: Can $H_n(q,k)$ be realized as an endomorphism algebra as above (or convolution algebra) in the general case (i.e. for any $q\in k$, an $e$-th primitive root of unity and $\operatorname{gcd}(e,p)=1$ )?
I don't know much about Hecke algebras, expecting your reply, thank you.
