An arithmetic highest weight theory?

Question

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.

Answer 1

Johnson, you have one of the foremost experts in the world on such matters (over general fields) just upstairs from your office. Make use of that.

Many of the basic constructions work for split groups over fields, but proving good properties (such as irreducibility and classification results) requires being over a field of characteristic 0. (Once constructions are made, to prove things one can extend scalars to an algebraic closure, or even reduce to the familiar case over $\mathbf{C}$ if so inclined, by the "Lefschetz Principle".) The Lie algebra is a good invariant (e.g., faithful!) over fields of characteristic 0, but even then it only retains at best information about groups up to isogeny. Another case where it gives a useful invariant is over $\mathbf{Z}/p\mathbf{Z}$-algebras where, together with the $p$-Lie algebra structure, it gives an equivalence with the category of finite locally free groups $G$ with vanishing relative Frobenius morphism such that the sheaf of invariant 1-forms is locally free over the base (loosely speaking because such vanishing allows one to get by with truncated exponential in degrees $< p$); this is explained in SGA3, VII$_{\rm{A}}$, 7.2, 7.4

More generally one cannot expect to get mileage out of the Lie algebra alone (but it is still perfectly useful via its role in classification by means of root systems, among other things).