The Torus Centralizing Subalgebra of $\text{Dist}(G_r)$

Let $G$ be a simple and simply connected algebraic group over a field of characteristic $p>0$, and $G_r$ its $r$-th Frobenius kernel. In this paper, we initiate a general study of $\text{Dist}(G_r{)}^T$, the subalgebra of $\text{Dist}(G_r)$ consisting of fixed points for the adjoint action of a maximal torus $T$ of $G$. We analyze the structure of this algebra, and classify its simple modules, which essentially are just the non-zero weight spaces of the simple $G_{r}T$-modules of $p^r$-restricted highest weight. Further connections between the representations of $\text{Dist}(G_r{)}^T$ and $G_{r}T$ are shown, demonstrating the potential usefulness of this algebra.