- #1

jdstokes

- 523

- 1

Suppose [itex]\mathfrak{g}[/itex] is the Lie alebra of G with maximal torus T and Cartan subalgebra [itex]\mathfrak{t}[/itex]. The weights are the (1-dimensional) irreducible represenations of T. If we restrict any representation [itex]\rho : G \to GL(V)[/itex] to T ([itex]\rho|_T : T \to GL(V)[/itex]) then we get a direct sum of weights [itex]\alpha_i[/itex]. If [itex]\rho[/itex] 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 [itex]\mathfrak{t}[/itex]?