Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in general, or for specific examples - about the ring of invariants $R_W:=k[W]^{SL_2}$.
One specific example that I'd like to know about is the case where $W$ is the space $S^d V$ of binary forms of degree $d$, or where $W$ is a direct sum of such spaces.
Let $D_W$ denote the multiset of the degrees of generators for $R_W$ (as an algebra over $k$). Apart from the calculation of $D_W$ and, ideally, explicit generators and relations for the examples from binary forms (for small degrees and specific values of $p$), I'd like to know the answer to general questions such as the following:-
For what values of $p$ is $D_{S^dV}$ the same as it would be in characteristic zero?