would like to briefly sketch for types $A_n$ and $C_n$ how can be derived via a sign-reversing involution.

\[ex:Weyl\] For ${\mathfrak{g}}=A_n$ the Weyl group $W$ is generated by the reflections $r_1,\ldots,r_n$ which act on ${\lambda}\in{\mathbb{Z}}^{n+1}$ as follows $$r_i({\lambda})=({\lambda}_1,\ldots,{\lambda}_{i+1},{\lambda}_i,\ldots,{\lambda}_{n+1}).$$ Hence on ${\mathbb{Z}}^{n+1}$ the Weyl group acts as the symmetric group $S_{n+1}$.

For ${\mathfrak{g}}=C_n$ the Weyl group is generated by $$\begin{aligned}
r_i({\lambda})&=({\lambda}_1,\ldots,{\lambda}_{i+1},{\lambda}_i,\ldots,{\lambda}_n) \qquad
 \text{for $1\le i<n$}\\
r_n({\lambda})&=({\lambda}_1,\ldots,{\lambda}_{n-1},-{\lambda}_n)
\end{aligned}$$ so that $W$ on ${\mathbb{Z}}^n$ acts by all permutations and sign changes.

Set $$\mathcal{S}=\{(\omega,b)\in W\times B \mid \omega({\mathrm{wt}}(b)+\rho)={\Lambda}+\rho\}$$ On the set $\mathcal{S}$ we define an involution $\Phi:\mathcal{S}\to\mathcal{S}$ with the properties that the fixed points are the pairs $(1,b)$ with $b\in {\mathcal{P}}'(B,{\Lambda})$ which recall are the paths underlying ${\overline{X}}(B,{\Lambda})$ (see ). Furthermore, if $\Phi(\omega,b)=(\omega',b')$ with $(\omega,b)\neq (\omega',b')$ then $\omega$ and $\omega'$ have opposite signs.

Let $\mathcal{S}_i$ for $i\in J$ be the set of pairs $(\omega,b)\in \mathcal{S}$ such that $\epsilon_i(b)>0$. Define $\Phi_i:\mathcal{S}_i\to\mathcal{S}_i$ by $\Phi_i(\omega,b)=(\omega r_i,s_i e_i(b))$. Define the set $\mathcal{S}'=\mathcal{S}-\{(1,b)\mid b\in{\mathcal{P}}'(B,{\Lambda})\}$ so that $\mathcal{S}'=\bigcup_{i\in J} \mathcal{S}_i$. Then $\Phi$ is given by $$\Phi(\omega,b)=\begin{cases}
 (\omega,b) & \text{if $(\omega,b)\in \mathcal{S}\setminus \mathcal{S}'$}\\
 \Phi_i(\omega,b) & \text{if $(\omega,b)\in \mathcal{S}'$ and $i=v(\omega,b)$}
\end{cases}$$ where $v$ is some functions $v:\mathcal{S}'\to J$. To show that $\Phi$ indeed exists and is an involution it needs to 