I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps group schemes) over a non-algebraically closed field (resp. a "nice" ring, say a Dedekind domain).
Are there analogous results to Lie's theorems in the case of algebraic groups (perhaps even arbitrary group schemes)? I am aware of Jantzen's book on representations of algebraic groups, but if I remember correctly, he does everything over an algebraically closed base field. I have not studied the book in detail to convince myself that the arguments there will carry over to the non-algebraically closed case.
I suppose the Borel-Bott-Weil-(Schmidt) construction of highest weights using sections of cohomology groups of line bundles may be generalized to a more arithmetic setting (as Jantzen has done in his book). Is there any progress in this direction beyond algebraic groups, say to include a "nice" class of group schemes? I am more curious of the case of classical groups.
Concerning more general group schemes, I have looked up parts of SGA3, but I did not find any clearly stated results connecting the Lie algebra of a group scheme (as defined there using universal properties) to the underlying group scheme.
A more general and more loaded question: to what extent is a smooth scheme determined by its tangent space at a distinguished point. I am aware of the notion of jet schemes, are there some important or at least neat results in this area anyone would like to share?
Thanks in advance.