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.