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

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