Understanding Weights & Roots: Why Does T Act Trivially?

[jdstokes]

I'm having trouble understanding the idea of a weight space.

Suppose \mathfrak{g} is the Lie alebra of G with maximal torus T and Cartan subalgebra \mathfrak{t}. The weights are the (1-dimensional) irreducible represenations of T. If we restrict any representation \rho : G \to GL(V) to T (\rho|_T : T \to GL(V)) then we get a direct sum of weights \alpha_i. If \rho is taken to be the adjoint representation, then the roots are defined to be the nontrivial weights of this rep.

My question concerns the trivial weights. Why exactly is it that T acts trivially on its own tangent space \mathfrak{t}?

 

[jdstokes]

I realized that this is pretty obvious since the adjoint action of T on itself is just identity because T is abelian. Thus for any t \in T, the linear transformation Ad|_T (t) : \mathfrak{g} \to \mathfrak{g} acts trivially on the Cartan subalgebra \mathfrak{h}. The T-action on the remainder of the Lie algebra, however may be non-trivial, which is where the nontrivial weights (ie roots) enter.