I've been studying algebraic groups and there is a confusion that I have been unable to resolve.
Let $G$ be a semisimple algebraic group and $T\subset G$ a maximal torus. If we let $T$ act on ${\rm Lie}(G)$ via the adjoint action we can see that $\mathfrak{g}$ decomposes into the direct sum of eigenspaces.
My question is this:
Are the roots of $G$ with respect to $T$ equal to the non trivial characters of $T$ which are also eigenvalues?
Any help would be greatly appreciated