5 expanded

This statement is false in general for algebraic groups. It's true in characteristic 0, but it is not in general true in positive characteristic. Instead, one has a weaker statement in positive characteristic (cf Proposition 4.20 on page 213 in Jantzen's "Algebraic Groups"):

Let $G$ be a reductive linear algebraic group over an algebraically closed field of positive characteristic $k$. Then $k[G]$ has an increasing filtration whose subquotients are of the form $H(\lambda) \otimes H(-w_0 \lambda)$, where $\lambda$ runs over the dominant weights for $G$ and the $H(\lambda)$ are the modules arising as global sections of line bundles on the flag variety of $G$ (the so-called costandard modules for $G$).

Moreover, this is true when $k[G]$ is considered as a $G\times G$-module.

Note that unlike in characteristic 0, these modules $V$ are not in general irreducible. (It's worth noting that the category of modules over a reductive algebraic group is not in general a semisimple category — this is only true in characteristic 0).

Let G $G$ be a reductive linear algebraic group over an algebraically closed field of positive characteristic . $k$. Then k[G] $k[G]$ has an increasing filtration whose subquotients are of the form H(\lambda) $H(\lambda) \otimes H(-w_0 \lambda), lambda)$, where \lambda $\lambda$ runs over the dominant weights for G $G$ and the H(\lambda) $H(\lambda)$ are the modules arising as global sections of line bundles on the flag variety of G $G$ (the so-called costandard modules for G). $G$).
Note that unlike in characteristic 0, these modules V $V$ are not in general irreducible. (It's worth noting that the category of modules over a reductive algebraic group is not in general a semisimple category -- this is only true in characteristic 0).