Let $G=GL\_n$ and $U\_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U\_-B$, where $B$ is the set of all upper triangular matrices. The group $U\_-$ acts on $G$ by left multiplication. Since $G = U\_-B$, we have an embedding $B \hookrightarrow U\_-B/U\_- = G/U\_-$. Therefore $\mathbb{C}[G/U\_-] \hookrightarrow \mathbb{C}[B]$. We have $\mathbb{C}[G/U\_-] = \oplus\_{\lambda} V\_{\lambda}$, where $\lambda$ is a dominant weight and $V\_{\lambda}$ is the highest weight module of $U(\mathfrak{g})$ with highest weight $\lambda$, where $\mathfrak{g}$ is the Lie algebra of $G$.


My question is: how to write $\mathbb{C}[G/U\_-]=\oplus\_{\lambda} V\_{\lambda}$ explicitly? 


For example, let $G = GL\_3$. Then 
\begin{align}
\mathbb{C}[G/U\_-] = \mathbb{C}[g\_{11}, g\_{12}, g\_{13}],
\end{align}
where $g\_{11}, g\_{12}, g\_{13}$ are coordinate functions which sends a matrix $x$ to matrix coefficients $x\_{11}, x\_{12}, x\_{13}$ respectively. How to write the $V\_{\lambda}$ in
\begin{align}
\mathbb{C}[G/U\_-] = \mathbb{C}[g\_{11}, g\_{12}, g\_{13}]=\oplus\_{\lambda} V\_{\lambda}
\end{align}
explicitly? Thank you very much.



How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?
The subspace $V\_\lambda$ is very easy to see. Since $U^-$ is normalized by the maximal torus $T$ there is an action of $T$ on $G/U^-$ on the right. This means that $\mathbb C[G/U^-]$ carries a multigrading and $V\_\lambda$ is just one of the multihomogeneous pieces.


The problem is now that $\mathbb C[G/U^-]$ is very complicated as a ring. The case $G=SL(2)$ is deceiving. Here $\mathbb C[G/U^-]$ is indeed a polynomial ring in two variables and the $V\_\lambda$ are the binary forms of degree $\lambda$. 


In general, $\mathbb C[G/U^-]$ is generated by the coordinate functions on all fundamental representations. The relations can be shown to be all quadratic, a generalization of the Plücker relations. For example for $G=GL(n)$ the ring is generated by all "left justified" $m\times m$-minors for $m=1,\ldots,n$.


So your description of $\mathbb C[G/U^-]$ for $G=GL(3)$ is incorrect. In fact $G/U^-$ is $6$-dimensional. The right hand side is the affine cone over a partial flag variety namely the projective plane, hence affine $3$-space.


The description of $\mathbb C[G/U^-]$ as a ring is precisely the purpose of standard monomial theory. There, one constructs a set of monomials which forms a $\mathbb C$-basis and describes multiplication by straightening laws. 


