Questions about fundamental representations of $SL_3/U$. Consider the group $SL_3$. Let $U$ be the subgroup of $SL_3$ consisting of all upper triangular unipotent matrices. Then the algebra $\mathbb{C}[SL_3/U]$ is generated by $a_{11}, a_{21}, a_{31}, a_{11}a_{22}-a_{12}a_{21}, a_{11}a_{32}-a_{12}a_{31}, a_{21}a_{32}-a_{22}a_{31}$. There are two fundamental representations of $SL_3/U$: $V_{\omega_1}$ and $V_{\omega_2}$. Do we have the following result:  the heighest weight vector of $V_{\omega_1}$ is $a_{11}$ and the heighest weight vector of $V_{\omega_2}$ is $a_{11} a_{22}-a_{12}a_{21}$? Are there some references about representation theory of $SL_n$ and $SL_n/U$? Thank you very much.

Given a complex semisimple group $G$ with Borel $B=TU$, where $T$ is a maximal torus and $U$ is the unipotent radical of $B$, the representation $\mathbf{C}[G/U]$ is a multiplicity-free direct sum over the irreducible finite dimensional representations of $G$. This is a consequence of the following algebraic version of the Peter-Weyl theorem:
$$\mathbf{C}[G] \cong \bigoplus V \otimes V^*,$$ in which the isomorphism is of $G$-bimodules and the sum runs over all finite dimensional irreducible $G$-modules $V$; in turn the algebraic Peter-Weyl theorem is a consequence of the fact that each $G$-module is semisimple and that $G$ acts locally finitely on $\mathbf{C}[G]$. Each $V^*$ is a highest weight (right) module, which means its space of $U$-invariants is one dimensional, implying the claim that $\mathbf{C}[G/U]$ is a multiplicity-free sum of all the finite dimensional irreducible $G$-modules.
For the particular case in your question, one needs precise definitions of the functions $a_{ij}$; I believe that your assertions are correct with a certain convention for the indexing. 
For more information, you should search the internet using the keyword basic affine space to refer to $G/U$.
