On the extensions of certain representations 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,where $q$ is a power of a prime number $p$. Let $\Bbbk$ be a field and we study the extensions of certain $\bk\bg$-modules in this paper. We show that the extensions of any modules in $\mathscr{O}(\bg)$ by a finite-dimensional $\bk\bg$-module is zero if $p\ne \op{char}\bk\ge5$ or $\op{char}\bk=0$, where $\mathscr{O}(\bg)$ is the principal representation category defined in \cite{D1}. We determine the necessary and sufficient condition for the vanishing of extensions between naive induced modules. As an application, we give the condition of the vanishing of extensions between simple modules in $\mathscr{O}({\bf G})$ for $\bg=SL_2(\bar{\mathbb{F}}_q)$.