Let $H,X,Y$ be the standard triple for $\mathfrak{sl}(2,\mathbb{C})$, let $\alpha$ be the root of $\mathfrak{t}=i\mathbb{R}H$ in $\mathfrak{g}_\mathbb{C}$. Denote the colection of weights of $\pi_*|t$ by $\Lambda(\pi_*)$.
Show that $\Lambda(\pi_*)\subset \frac{1}{2}\mathbb{Z}\alpha$
I'm new to roots and weights so I a little unsure about this. I know that weights are given by $\lambda$ where $V_\lambda=\{v\in V | Hv=\lambda v\}$ and the collection of weights is $\Lambda(\pi)=\{\lambda| V_\lambda\neq0\}$. Roots are the weights for the the adjoint representation which are different from zero.
We must show that $V_{\lambda}$ is only non empty when $\lambda=n\alpha/2$. I must be overlooking something as it seems it is never the case that $Hv= \lambda v$?