Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I_G$ is the set of simple roots. Let $P$ be a standard parabolic subgroup, $M$ be its Levi containing $T$ and let $I_M$ be the set of simple roots of $M$ (with the natural choice of Borel of $M$). Let $\Lambda_G$ be the weight lattice of G. We define $\Lambda_{G,P}:=\frac{\Lambda_G}{\text{span of $\alpha_i$, $i\in I_M$}}$.

> Is $\Lambda_{G,P}=X(Z(M)^0)$, where $Z(M)^0$ is the component of $Z(M)$ (
the center of $M$)  containing the identity and $X(Z(M)^0)$ is the character group of the
> torus $Z(M)^0$?