= Solution
The <weight lattice> is
$$
X=\{\lambda\in\mathfrak t^*:
\langle\lambda,\alpha^\vee\rangle\in\mathbb Z
\text{ for every }\alpha\in\Phi\}.
$$
For each root $\alpha$, restrict $V$ to the subalgebra
$$
\mathfrak m_\alpha=\operatorname{span}\{e_\alpha,h_\alpha,f_\alpha\}
\cong\mathfrak{sl}_2.
$$
If $V_\lambda\ne0$, the $\mathfrak{sl}_2$ classification shows that the $h_\alpha$-eigenvalue $\lambda(h_\alpha)=\langle\lambda,\alpha^\vee\rangle$ is an integer. Hence $\lambda\in X$.
The same classification makes every $\alpha$-string symmetric under
$$
\lambda\longmapsto
\lambda-\langle\lambda,\alpha^\vee\rangle\alpha=s_\alpha\lambda
$$
and preserves weight multiplicity. Since the <Weyl group> is generated by these simple reflections,
$$
\boxed{V_{w\lambda}\ne0\quad\text{and}\quad
\dim V_{w\lambda}=\dim V_\lambda
\quad(w\in W)}.
$$
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