Any nontrivial torus is contained in a maximal torus (so that, in particular, contains a maximal torus).
Optional: A torus is contained in a maximal torus
Any torus is contained in a maximal torus
This video is optional.
We have so far proved that any compact matrix group contains a nontrivial torus. We will now show that:
Suppose, for a contradiction, that we have a torus not contained in a maximal torus. Then cannot itself be maximal (it's contained in itself), so it must be contained in a strictly bigger torus , and is also not allowed to be maximal. Therefore is contained in a strictly bigger non-maximal torus . Continuing in this manner, I can construct a sequence of nested tori each strictly included in the next.
We now pass to the level of Lie algebras: These are all subspaces of , which is a finite-dimensional vector space. Therefore this sequence of Lie algebras must "stabilise" at some point, that is there exists an such that for all .
We saw earlier that a torus is exp of its Lie algebra, so for all . But then for all , and this is a contradiction because is supposed to be a strict inclusion.
Outlook
Now we have found a maximal torus ( ), we will use it in the following way. Given a (smooth, complex) representation , we can restrict to get a representation of our maximal torus. Because , we deduce that where Here, .
The bigger our torus, the larger is, so the more integers we have to split up our weight spaces.
There is a more sophisticated way to think about . We can think of as the linear function of the s. The s are coordinates on a certain vector space. This is not quite the Lie algebra of , because the Lie algebra of consists of -tuples of imaginary numbers (just like how ). Instead, the vector lives in . This means that lives most naturally in (where means the dual space of linear functions). We will discuss this more formally when we talk about the Killing form.