Good l-filtrations for q-GL_3(k)

Abstract

Let $k$ be an algebraically closed field of characteristic $p$, possibly zero, and $G=q$-$\GL_3(k)$, the quantum group of three by three matrices as defined by Dipper and Donkin. We may also take $G$ to be $\GL_3(k)$. 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, $\hat{Z}(λ)$, for the infinitesimal group $G_1B$. We induce this structure to $G$ to obtain a good $l$-filtration of certain induced modules, $\nabla(λ)$, for $G$. We also determine the homomorphisms between induced modules for $G$.