Let $G$ be a semi simple algebraic group, $B \subset G$ is a Borel subgroup and $U \subset B$ is the unipotent radical of $B$. We can consider the variety $G/U$. Let us also denote $\overline{G/U}:=\operatorname{Spec}(\mathbb{C}[G/U])$. It is known that the natural morphism $G/U \rightarrow \overline{G/U}$ is an open embedding. Let $\partial{G/U}$ be the boundary of $G/U$ inside $\overline{G/U}$. Note now that $\mathbb{C}[G/U]=\bigoplus_{\mu} V(\mu)$, where the sum runs through dominant characters $\mu$ of $G$ (we fix some maximal torus $T \subset B$, here $V(\mu)$ is the irreducible representation of $G$ with highest weight $\mu$.

Claim: the ideal of $\partial{G/U} \subset \overline{G/U}$ is generated by $V(\mu)$ with $\mu$ being regular (strictly dominant). How to prove this claim? Maybe there are any references?

Let $G$ be a semi simple algebraic group, $B \subset G$ is a Borel subgroup and $U \subset B$ is the unipotent radical of $B$. We can consider the variety $G/U$. Let us also denote $\overline{G/U}:=\operatorname{Spec}(\mathbb{C}[G/U])$. It is known that the natural morphism $G/U \rightarrow \overline{G/U}$ is an open embedding. Let $\partial{G/U}$ be the boundary of $G/U$ inside $\overline{G/U}$. Note now that $\mathbb{C}[G/U]=\bigoplus_{\mu} V(\mu)$, where the sum runs through dominant characters $\mu$ of $G$ (we fix some maximal torus $T \subset B$, here $V(\mu)$ is the irreducible representation of $G$ with highest weight $\mu$).

Claim: the ideal of $\partial{G/U} \subset \overline{G/U}$ is generated by $V(\mu)$ with $\mu$ being regular (strictly dominant). How to prove this claim? Maybe there are any references?