Verma Modules for Restricted Quantum Groups at a Fourth Root of Unity

For a semisimple Lie algebra $\mathfrak{g}$ of rank $n$, let $\overline{U}_\zeta(\mathfrak{g})$ be the restricted quantum group of $\mathfrak{g}$ at a primitive fourth root of unity. This quantum group admits a natural Borel-induced representation $V({\boldsymbol{t}})$, with ${\boldsymbol{t}}\in(\mathbb{C}^\times)^n$ determined by a character on the Cartan subalgebra. Ohtsuki showed that for $\mathfrak{g}=\mathfrak{sl}_2$, the braid group representation determined by tensor powers of $V({\boldsymbol{t}})$ is the exterior algebra of the Burau representation. In this paper, we generalize the tensor decomposition of $V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})$ used in Ohtsuki's proof to any semisimple $\mathfrak{g}$. Upon specializing to the $\mathfrak{sl}_3$ case, we describe all projective covers of $V({\boldsymbol{t}})$ in terms of induced representations. The above decomposition formula for $V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})$ is then extended to more general $\boldsymbol{t}$ and $\boldsymbol{s}$ where these projective covers occur as indecomposable summands. We also define a stratification of $(\mathbb{C}^\times)^{4}$ whose points $({\boldsymbol{t}},{\boldsymbol{s}})$ in the lower strata are associated with representations $V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})$ that do not have a homogeneous cyclic generator. With this information, we characterize under what conditions the isomorphism ${V({\boldsymbol{t}})\otimes V({\boldsymbol{s}})\cong V({\boldsymbol{\lambda t}})\otimes V({\boldsymbol{\lambda^{-1} s}})}$ holds.