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Kernel of the map $mathbb{C}[G]^U to mathbb{C}[U^+]$

MathOverflow Asked by Jianrong Li on January 13, 2021

$DeclareMathOperator{SL}{operatorname{SL}}$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.

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