Remark 2

Let $\Theta (G)$ be the algebra of continuous representative functions on G with values in $\mathbf{C}($Algebra, Chap. VIII). Let G operate on $\Theta (G)$ by left and right translations. For each $\lambda \in X_{++}$, let $(V_{\lambda}, \tau_{\lambda})$ be an irreducible representation of G of highest weight $\lambda$ (Th. 1), and $(V^*_{\lambda},\check{\tau}_{\lambda})$ the contragredient representation (Chap. III, §3, no. 11); by Spectral Theories, the representation of $G\times G$ on $\Theta (G)$ is isomorphic to the direct sum of the representations $(V_{\lambda}\otimes V^*_{\lambda}, \tau_{\lambda}\otimes \check{\tau}_{\lambda})$ for all $\lambda$ in $X_{++}$. Remark 1 now implies the following statement: Let $\lambda \in X_{++}$, and let $E_{\lambda}$ be the vector subspace of $\Theta (G)$ consisting of the continuous representative functions $f$ on G such that $f(gt) =\lambda (t)^{-1}f(g)$ for all $g\in G$ and all $t\in T$, and such that $f*x= 0$ for all $x\in \mathfrak{n}_-=\bigoplus_{\alpha <0}\mathfrak{g}^{\alpha}$.

Then $E_{\lambda}$ is stable under left translations, and the representation of G on $E_{\lambda}$ by left translations is irreducible of highest weight $\lambda$.