Why are root data a natural candidate for classifying connected reductive groups?

Question

For the purpose of this question, you may assume that we are working over the complex numbers.

Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the Lie algebra $\mathfrak{g}$ of $G$. One can use this action to define the root datum, which in turn is invariant of the choice of $T$, and use it to classify connected reductive groups.

The action of $T$ on $\mathfrak{g}$ is nice in that it has more information than just the action of $T$ on the Lie algebra $\mathfrak{t}$ of $T$, and yet is simple enough so as to decompose into one dimensional weight spaces. But that is a far cry from saying that this is a natural action to consider when trying to classify connected reductive groups!

Is there a deep reason that root data, or more generally the action of a maximal torus on the Lie algebra of $G$, is a natural thing to consider? Does it correspond to some cohomological invariant? Does it arise naturally? Or is this entire theory a fluke?

The proof does not seem to bring much insight into this story, in that it boils down to a series of reductions, which reinforces for me the suspicion that root data are not in and of themselves natural, but rather that this was a guess for a way to classify connected reductive groups that just happened to work out...

Answer 1

I can't give you a very deep reason for why root data appear in this context (because, let's face it, root systems spring out everywhere), but there are some very elementary reasons to why the action in question is very natural with regard to the classification.

Let me start with the following two considerations:

• When one tries to distinguish between the two objects, one usually looks for some simple properties which differ between them. For example, to show that two abstract groups are not isomorphic, one starts by comparing their orders, and procede by comparing the number of elements of a given order in each of them or which subgroups are there and how they fit together.
• In a semisimple Lie algebra $L$ there is a Jordan decomposition, which tells that every elements $x$ is the sum of an $\operatorname{ad}$-semisimple element $x_s$ and an $\operatorname{ad}$-nilpotent part $x_n$. And there is a subalgebra consisting of semisimple elements (otherwise $L$ itself is nilpotent by Engel theorem). Such subalgebras are called toric, and it turns out they are always abelian. Thus when considered in their adjoint repesentation, the elements of a toric subalgebra form a commuting family of semisimple endomorphisms of $L$, hence are simultaneously diagonalizable, which is equivalent to $L$ decomposing into the direct sum of its weight subspaces, which gives rise to the root system.

So combining these two considerations, to distinguish (and ultimately classify) semisimple Lie algebras we essentially take the simplest type of elements of $L$ (the semisimple ones) and look at how we can fit them together in $L$ (so that they from a subalgebra, and a maximal such).

This looks somewhat abstract, but really just mimics what can be easily seen in the examples, namely, in the classical semisimple Lie algebras. The standard constructions in their minimal representations are equipped with some very simple bases (for example, what comes first to mind for $\mathfrak{sl}_n$?), and there is a very natural maximal toric subalgebra $H$, namely, the diagonal matrices. The non-zero-weight subspaces are the spans of individual off-diagonal basis elements, and the root system captures their configuration.

Now getting back to algebraic groups, they can be roughly classified by the isomorphism classes of the corresponding Lie algebras, but additional information is needed to account for the center. The center sits inside the torus, so to incorporate this missing data into the rough classification one translates the adjoint action of the corresponding toric subalgebra to the adjoint action of the torus.