What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, singularities...?
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The Isogeny Theorem does this for all isogenies between split reductive groups over any field (including non-central isogenies and exceptional isogenies in small characteristics) via the notion of "$p$-morphism" between root data ($p \ge 1$). Is there a motivating situation to say more? For non-split groups the "anisotropic kernel" is invisible to root systems. For split semisimple groups in positive characteristic the homomorphisms into ${\rm{GL}}_n$ seem too rich to be encoded in terms of combinatorial information in a functorial way (e.g., there is a "highest weight" theory, but dimensions of irreducibles are mysterious, and semisimplicity fails badly). Complexities due to working over a ground field that isn't algebraically closed (even char. 0) seem huge, such as for field of definition of representations, much as we know very little about ${\rm{H}}^1(k,G)$ for general $k$ and many split $G$. |
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