For finite groups there is an isomorphism $$k[G]\cong \bigoplus\limits_{V -irrep}\mathrm{End}(V)$$ compatible with the group action. Can this fact be generalized to the case of linear algebraic groups?(there $k[G]$ is understood as the ring of regular functions).
In particular, I am interested in the case $G=SL_2$ so the RHS is equal to $$\bigoplus_n\mathrm{End\,Sym^n}k^2$$