Good l-filtrations for q-GL_3(k)

Let $k$ be an algebraically closed field of characteristic $p$, possibly zero, and $G=q$-${\text{GL}}_3(k)$, the quantum group of three by three matrices as defined by Dipper and Donkin. We may also take $G$ to be ${\text{GL}}_3(k)$. We first determine the extensions between simple $G$-modules for both $G$ and $G_1$, the first Frobneius kernel of $G$. We then determine the submodule structure of certain induced modules, $\widehat{Z}(\lambda )$, for the infinitesimal group $G_{1}B$. We induce this structure to $G$ to obtain a good $l$-filtration of certain induced modules, $\mathrm{\nabla }(\lambda )$, for $G$. We also determine the homomorphisms between induced modules for $G$.