$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $G=\SL_k$ be the special linear group, $U$ the unipotent subgroup consisting of all lower unipotent triangular matrices, $U^+$ the unipotent subgroup consisting of all upper triangular matrices.
Denote by $\mathbb{C}[\SL_k]^U$ the $U$-invariant functions on $\SL_k$. Consider the map $\varphi: \mathbb{C}[\SL_k]^U \to \mathbb{C}[U^+]$ defined by restricting $U$-invariant functions on $\SL_k$ to the subgroup $U^+$. As shown in the book by Fomin–Williams–Zelevinsky, Remark 6.5.7, this map is onto.
What is the kernel of $\varphi$? Is it the ideal generated by leading principal minors?
Denote by $\widetilde{\mathbb{C}[\SL_k]^U}$ the quotient of $\mathbb{C}[\SL_k]^U$ by the ideal generated by leading principal minors. Is it true that $\widetilde{\mathbb{C}[\SL_k]^U}$ is isomorphic to $\mathbb{C}[U^+]$ as algebras?
Thank you very much.
