I'll use "affinization" to describe the natural map of schemes $X \rightarrow \text{Spec}(\Gamma(X, \mathcal{O}\_X))$. For quasi-affine varieties $X$ this is an open embedding.


Let $G$ be a reductive linear group and $\omega\_1, \ldots, \omega\_n$ a choice of fundamental weights. We can get a "Plucker embedding" $G/B \hookrightarrow \Pi\_{i=1}^n \mathbb{P}(V\_{\omega\_i})$ which sends a Borel to its eigenspace in each representation. We can also take an embedding:
$$G/N \hookrightarrow \Pi\_{i=1}^n (V\_{\omega\_i} - 0) \hookrightarrow \Pi\_{i=1}^n V\_{\omega\_i}$$
This is an open embedding, so $G/N$ is quasiaffine.


The question is: what is the affinization of $G/N$? I have a guess, which is that it is the closure of the image under the embedding. Is this true? If it's true, why (some general fact about the affinization of quasiaffine varieties, or a more specific argument)?


Also, there should be a description of this set that goes something like: collections of vectors $(v\_1, \ldots, v\_n) \in V\_{\omega\_1} \times \cdots \times V\_{\omega\_n}$ such that the $v\_i$ are eigenvectors of the same Borel, or zero. I don't really have an argument for this either, though.



"Plucker" embedding of G/N, for reductive group G, affinization of quasiaffine varieties
The functions on $G/N$ are a ring of the form $\oplus\_{\lambda}V\_{\lambda}^\star$ where $\lambda$ ranges over all dominant integral weights. For me, $V\_\lambda$ has a fixed highest weight vector $v\_\lambda$, and the map is given by $v\mapsto \langle v,gv\_\lambda\rangle$ as a function of $g$. The multiplication is given by Cartan product (the unique projection $V\_{\lambda\_1}^\star\otimes V\_{\lambda\_2}^\star\to V\_{\lambda\_1+\lambda\_2}^\star$ which sends $v\_{\lambda\_1+\lambda\_2}\to v\_{\lambda\_1}\otimes v\_{\lambda\_2}$).


In particular, $\oplus V\_{\omega\_i}^\star$ are generators for this ring, showing the affinization embeds in the corresponding affine space $\oplus V\_{\omega\_i}$ as the closure of the image of $G/N$. Furthermore, the form of the map shows that the coset $g$ is sent to $(gv\_{\omega\_1},\dots, gv\_{\omega\_n})$, so indeed the affinization is the closure of this set.


