"Plucker" embedding of G/N, for reductive group G, affinization of quasiaffine varieties

Question

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.

Answer 1

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.