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.