The block decomposition of the principal representation category of reductive algebraic groups with Frobenius maps

Let $\mathbf{G}$ be a connected reductive algebraic group defined over the finite field ${\mathbb{F}}_q$ with $q$ elements. Let $\mathbb{k}$ be a field such that $\text{op}char\mathbb{k}\mathrm{\ne }\text{op}char{\mathbb{F}}_q$. In this paper, we study the extensions of simple modules (over $\mathbb{k}$) in the principal representation category $\mathcal{O}(\mathbf{G})$ which is defined in \cite{D1}. In particular, we get the block decomposition of $\mathcal{O}(\mathbf{G})$, which is parameterized by the central characters of $\mathbf{G}$.