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Let $G=GL_n$ and $N$ its unipotent subgroup consisting of all upper unipotent triangular matrices. Do we have $\mathbb{C}[G/N] = \mathbb{C}[G]^N$? Here $\mathbb{C}[G/N]$ is the coordinate ring of $G/N$ and $\mathbb{C}[G]^N$ is the set of functions in $\mathbb{C}[G]$ which are invariant under the action of $N$, where the action of $N$ on $G$ is given by right multiplication. Thank you very much.

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The question as stated is not well-defined because you will have to say how you consider G/N as an algebraic variety. There is however a canonical way of doing this, the buzzwords are "generalized flag variety" and "Plucker embedding". The coordinate ring of G/N are then the Plucker coordinates and these are indeed isomorphic to the N-invariant elements of C[G].

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