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Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are assigned orientations. Let $p$ be a vertex of $\vec{\Gamma}$ attached to a leaf $q$ and let $m= m_{p,q}$ be the weight of the corresponding edge. Let $\mathring{\vec{\Gamma}}$ be the induced coxeter diagram obtained by removing the leaf $q$.

Let $G$ and $\mathring{G}$ be the Kac-Moody groups associated to $\vec{\Gamma}$ and $\mathring{\vec{\Gamma}}$ respectively. Choose Borel subgroups $B$ and $\mathring{B}$ of $G$ and $\mathring{G}$ and let $N$ and $\mathring{N}$ denote their associated unipotent radicals.

Question: Is there a representation $\mathcal{H}_{p,m}$ of $\mathring{N}$ such that $N \cong \mathring{N} \rtimes \mathcal{H}_{p,m}$ ?

Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are assigned orientations. Let $p$ be a vertex of $\vec{\Gamma}$ attached to a leaf $q$ and let $m= m_{p,q}$ be the weight of the corresponding edge. Let $\mathring{\vec{\Gamma}}$ be the induced coxeter diagram obtained by removing the leaf $q$.

Let $G$ and $\mathring{G}$ be the Kac-Moody groups associated to $\vec{\Gamma}$ and $\mathring{\vec{\Gamma}}$ respectively. Choose Borel subgroups $B$ and $\mathring{B}$ of $G$ and $\mathring{G}$ and let $N$ and $\mathring{N}$ denote their associated unipotent radicals.

Question: Is there a representation $\mathcal{H}_{p,m}$ of $\mathring{N}$ such that $N \cong \mathcal{H}_{p,m} \rtimes \mathring{N}$ ?

Consider a coxeter diagram $\Gamma$, i.e. a finite graph whose edges are decorated by one of $\big\{3,4,6\big\}$. Let $p$ be a vertex of $\Gamma$ attached to a leaf $q$ and let $m= m_{p,q}$ be the weight of the corresponding edge. Let $\mathring{\Gamma}$ be the induced coxeter diagram obtained by removing the leaf $q$.

Let $G$ and $\mathring{G}$ be the Kac-Moody groups associated to $\Gamma$ and $\mathring{\Gamma}$ respectively. Choose Borel subgroups $B$ and $\mathring{B}$ of $G$ and $\mathring{G}$ and let $N$ and $\mathring{N}$ denote their associated unipotent radicals.

Question: Is there a representation $\mathcal{H}_{p,m}$ of $\mathring{N}$ such that $N \cong \mathring{N} \rtimes \mathcal{H}_{p,m}$ ?

Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are assigned orientations. Let $p$ be a vertex of $\vec{\Gamma}$ attached to a leaf $q$ and let $m= m_{p,q}$ be the weight of the corresponding edge. Let $\mathring{\vec{\Gamma}}$ be the induced coxeter diagram obtained by removing the leaf $q$.

Let $G$ and $\mathring{G}$ be the Kac-Moody groups associated to $\vec{\Gamma}$ and $\mathring{\vec{\Gamma}}$ respectively. Choose Borel subgroups $B$ and $\mathring{B}$ of $G$ and $\mathring{G}$ and let $N$ and $\mathring{N}$ denote their associated unipotent radicals.

Question: Is there a representation $\mathcal{H}_{p,m}$ of $\mathring{N}$ such that $N \cong \mathring{N} \rtimes \mathcal{H}_{p,m}$ ?