To my understanding there is an algebraic version of Peter-Weyl that holds in characteristic $0$ that says for any reductive group $G$ one has that:
$$k[G]=\bigoplus V\otimes V^*$$
as a $G\times G$-representation, where the sum runs over all irreducible representations of $G$, $V$.
I've heard that such a theorem may hold in positive characteristic, though maybe one needs to adjust to $V$ to a different class of $G$-modules like Weyl modules. My concern is here that in positive characteristic many results need modified as reductive groups aren't linearly reductive (reps don't decompose into irreps). The proofs I've seen of algebraic Peter-Weyl use linear reductiveness.
Does anyone have a reference for this? Is this true/ is there any way to decompose the action of $G$ on it's coordinate ring $k[G]$ when the base field $k$ is positive characteristic? (I'm more than happy to assume $k$ is algebraically closed).
Edit: I will mention that the general definition of reductive here is that $G$ is a linear algebraic group with trivial unipotent radical.
Best Answer
The answer to this question is that, no, such a decomposition doesn't quite work, but there is a similar result described here: mathoverflow.net/questions/3474/decomposition-of-kg if you look at Chuck Hague's answer. Thanks to the commented who mentioned this. One can interpret this as having a filtration of $k[G]$ as a $G\times G$ rep where the quotients are given by the simples and cosimples in the category of representations of $G$. Forgive me if this is fuzzy this isn't my area of expertise.