Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat decomposition $G = \cup\_{w \in W} BwB = \cup\_{w \in W} BwU$. Let $U$ act on $G$ by right multiplication. Then we have an embedding $B \hookrightarrow G/U$. Therefore there is an embedding $ℂ[G/U]↪ℂ[B]$. 


My question is: how to write the map $ℂ[G/U]↪ℂ[B]$ explicitly? For example, let $G=GL\_2$, $B$ the subgroup of all upper triangular matrices, and $U$ the subgroup consisting of all upper triangular matrices. Then we have 
$$ ℂ[G/U]=ℂ[g\_{11},g\_{21},g\_{11}g\_{22}−g\_{12}g\_{21}]$$ and $$ℂ[B]=ℂ[b\_{11},b\_{12},b\_{22}].$$ 
What are the images of $g\_{11},g\_{21},g\_{11}g\_{22}−g\_{12}g\_{21}$? Thank you very much.



How to write the map $ℂ[G/U]↪ℂ[B]$ explicitly?
First of all, for $GL(2)$ you have to add $(g\_{11}g\_{22}-g\_{12}g\_{21})^{-1}$ to your generators. Same for $B$: also $b\_{11}^{-1}$ and $b\_{22}^{-1}$ are generators.


The map $B\to G/U$ is $b\mapsto bw\_0U$ where $w\_0$ is the longest Weyl group element. The map on rings is just restriction. Thus $f(gU)$ is mapped to $f(bw\_0)$. For $GL(2)$ this reads
$$
g\_{11}\mapsto b\_{12},g\_{21}\mapsto b\_{22},g\_{11}g\_{22}-g\_{12}g\_{21}\mapsto -b\_{11}b\_{22}
$$
where I chose $w\_0=\begin{pmatrix}0&1\\1&0\end{pmatrix}$.


