Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient stack. I was wondering what can be said about the cohomology $R\Gamma([V/G], \mathcal{O}_{[V/G]})$ (or in other words about the derived $G$-invariants of the ring of functions $\Gamma(V)$)? Namely I would like to know how much does it differ from the usual invariants $\Gamma(V)^G$ in terms of the representation $V$ (plus how bad $p$ is for $G$).

The problem is in positive characteristic $G$ is usually no more linearly reductive and I do not see any easy way to compute the higher cohomologies.

When $G$ is equal to $GL_n$ I would be particularly interested in the case when $V$ is a direct sum of copies of the tautological $n$-dimensional representation (and so the question is how $R\Gamma$ depends on $p$). In general any references/comments about that will be helpful.