Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center. Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$.

Let $\chi : T(\mathbf{Q}_p) \to \mathbf{Z}_p^\times$ be a continuous character and assume $\chi \circ \alpha^\vee \neq 1$ for all $\alpha \in R$.

**Question:** Do we have $w(\chi)=\chi$ if and only if $w=1$ ?

This is true for $\mathrm{GL}_n$ or $\mathrm{GSp}_{2n}$ and also for unramified characters, but is it in general ?

Remark: If the center of $G$ is not connected there are counterexamples, e.g. $G=\mathrm{SL}_2$ and $\chi : \mathrm{diag}(x,x^{-1}) \mapsto (-1)^{\mathrm{ord}_p(x)}$.

(Edit) Remark: If the center of $G$ is connected and $\alpha \in R$, then $\chi \circ \alpha^\vee \neq 1 \Leftrightarrow s_\alpha(\chi) \neq \chi$.

**Edit / Answer:** The connectedness of the center is not a sufficient condition. There is a counterexample with $G_2$ : the longest element of its Weyl group $w_0$ acts on $T(\mathbf{Q}_p)$ by $w_0(t)=t^{-1}$ and one can construct a character $\chi$ such that $\chi \circ \alpha^\vee \neq 1$ for all $\alpha \in R$ and $w_0(\chi)=\chi$ as follow. Assume $p \neq 2$ so that $\mathbf{Q}_p^\times \cong \mathbf{Z} \times \left( \mathbf{Z}/p\mathbf{Z} \right)^\times \times \mathbf{Z}_p$ and let $\chi_1 : \mathbf{Q}_p^\times \twoheadrightarrow \mathbf{Z} \to \{\pm 1\}$ and $\chi_2 : \mathbf{Q}_p^\times \twoheadrightarrow \left( \mathbf{Z}/p\mathbf{Z} \right)^\times \to \{\pm 1\}$, then define $\chi : T(\mathbf{Q}_p) \cong \mathbf{Q}_p^\times \times \mathbf{Q}_p^\times \to \{\pm 1\}$ by $\chi(t_1,t_2)=\chi_1(t_1)\chi_2(t_2)$.