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.