a_{3}-d,a_{4}\ldots,a_{n}\right)$$ where $d=a_{1}+a_{2}+a_{3}-a_{0}$, while the reflexion $r_{\alpha_{i}}$, $i\geq 1$, permutes the coefficients $a_{i}$ and $a_{i+1}$.

The key ingredient to understand the action of $D_{K}(1,n)$ on the symplectic cone $\mathcal{C}_{K}$ is the notion of a *reduced class*:

\[def:ReducedClasses\] Let $k\geq 3$. A class $A=a_{0}L-\sum_{i} a_{i}E_{i}$ is said to be reduced with respect to the basis $\{L, E_{1},\ldots,E_{k}\}$ if $a_{1}\geq a_{2}\geq\cdots\geq a_{k}\geq 0$ and $a_{0} \geq a_{1}+a_{2}+a_{3}$.

\[thm:Diff+ReducedClasses\] Assume $n\geq 3$.

1.  [(]{}see[ [@Li-Li-Diff], Theorem 3.1)]{} The group $D(1,n)$ is generated by the reflections $\{r_{L}, r_{E_{1}},
        r_{\alpha_{0}},\ldots,r_{\alpha_{n}}\}$. In particular, it follows that the group $D_{K}(1,n)$ is generated by the reflections $\{r_{\alpha_{0}},\ldots,r_{\alpha_{n}}\}$.

2.  [(]{}see [[@Li-Li-SymplecticGenus], Theorem D and [@Li-Liu-Uniqueness], Theorem 1)]{} The group $D_{K}(1,n)$ acts transitively on ${\mathcal{E}}_{K}$.

3.  [(]{}see [[@Gao], Proposition 2.2)]{} The orbit of an element $A\in{\mathcal{P}}$ under the action of $D(1,n)$ contains a unique reduced class.

4.  [(]{}see [[@Li-Liu-Uniqueness], Proposition 4.9 (3))]{} A reduced class $A=a_{0}L-\sum_{i}a_{i}E_{i}$ belongs to $\mathcal{C}_{K}$ if and only if $a_{i}>0$ for all $i$.

Combining Theorem \[thm:CharacterizationEK\] with Theorem \[thm:Diff+ReducedClasses\] (2), we have

\[cor:FundamentalDomain\] Let $C_{0}$ denote the set of reduced classes $a_{0}L-\sum_{i}a_{i}E_{i}$ with $a_{i}>0$, $\forall i$.

1.  A class $A\in{\mathcal{P}}_{+}$ belongs to $\mathcal{C}_{K}$ if and only if its orbit under $D_{K}(1,n)$ only contains classes $A=a_{0}L-\sum_{i}a_{i}E_{i}$ with 