Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+$ for the set of positive roots in $\Phi$.
Fix a subset of simple roots $I$ and let $W_I$ be the corresponding standard parabolic subgroup of $W$, with longest element $w_I$ and root system $\Phi_I\subseteq \Phi$. Let $\Phi_I^+:=\Phi_I\cap\Phi^+$.
Define $ \Lambda^+_I := \{\nu \in \mathfrak{h}^* : \langle\nu,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}. $
Consider $\lambda \in \Lambda^+_I$ and assume $\lambda$ is integral. We define $ {}^IW := \{w\in W: w<s_\alpha w \ \text{for all }\alpha\in I\}, $ where $<$ is the Bruhat ordering on $W$.
Denote by $\Delta$ the simple system corresponding to the positive system $\Phi^+$ in $\Phi$. The orbit $W\cdot\lambda$ contains a unique $\mu\in\mathfrak{h}^*$ that is antidominant in the sense that $\langle \mu+\rho,\alpha^{\lor}\rangle\not\in\mathbb{Z}^{>0}$ for all $\alpha\in \Phi^+$, where $\rho = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$.
The set of singular simple roots associated to $\mu$ in $\Delta$ is defined by $ {\Sigma_\mu} : = \{\alpha\in \Delta: \langle\mu+\rho,\alpha^\lor\rangle=0\}. $
The subgroup $W_{\Sigma_\mu} := \{w\in W: w(\mu+\rho)=\mu+\rho\}\subseteq W$ is then the isotropy group of $\mu$.
Let $ {}^IW^{\Sigma_\mu} : = \{w\in {}^IW: w<ws_\alpha\in {}^IW\ \text{for all }\alpha\in {\Sigma_\mu}\}, $ where $<$ is the Bruhat ordering on $W$.
Let $M_I(\eta)$ be the parabolic Verma module with highest weight $\eta$ and $L(\eta)$ be its unique simple quotient.
From section 8.3 of KOSTANT MODULES IN BLOCKS OF CATEGORY $O_S$:
For $x<w$ in ${}^IW^{\Sigma_\mu}$. Define $c_I(x,w)=\dim \mathrm{Ext}^1_{\mathcal{O}^\mathfrak{p}}\left(M_I(w_Ix\cdot\mu),L(w_Iw\cdot\mu)\right)$.
For $x, w \in {}^IW^{\Sigma_\mu}$, write $x \to_c w$ if $x < w$ in the Bruhat ordering, $c_I(x, w) \neq 0$ and there is no $x < z < w$ in ${}^IW^{\Sigma_\mu}$ with $c_I(z, w) \neq 0$. Then define $\le_c$ as the ordering on ${}^IW^{\Sigma_\mu}$ generated by the covering relations $x \to_c w$. We call this ordering on ${}^IW^{\Sigma_\mu}$ the $\mathrm{Ext}^1$-ordering.
I would like to know whether the sentences "For any $w\in {}^IW^{\Sigma_\mu}-\{e\}$, there exists $x\in {}^IW^{\Sigma_\mu}$ such that $x<w$, where $<$ is the Bruhat ordering on $W$." and "For any $w\in {}^IW^{\Sigma_\mu}-\{e\}$, there exists $x\in {}^IW^{\Sigma_\mu}$ such that $x<_cw$." are true or not?
