Is there an example of an algebraic group $G$ with maximal torus $T$ and Weyl group $W$ of type $B_n$, (a specific $n$ is fine), and a character $\lambda\in X=\hom(T,\mathbb{G}_m)$ such that $\operatorname{stab}_W(\lambda)\cong\mathbb{Z}/(2)\times\mathbb{Z}/(2)$?
The stabilizer being with respect to the action of the Weyl group on $X$ given by $^w\chi(t)=\chi(t^w)$, for $\chi\in X$ and $t\in T$. Thank you.
Let $\{\alpha_1, \ldots, \alpha_l\}$ be a base for the root system with corresponding fundamental dominant weights $\omega_1, \ldots, \omega_l$.
Write a dominant weight $\lambda$ as a sum of fundamental dominant weights, say $\lambda = \sum_{i = 1}^l a_i \omega_i$ where $a_i \in \mathbb{Z}_{\geq 0}$. Then we know (see for example Humphreys' book on Lie algebras) that $\operatorname{Stab}_W(\lambda)$ is generated by the simple reflections $\sigma_{\alpha_i}$ with $a_i = 0$. Now it should be easy for you to find many examples of $\lambda$ with $\operatorname{Stab}_W(\lambda) \cong \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}$.