I am studying Humphrey's book on BGG Category $\mathcal{O}$ and have been trying to understand the importance of Shapovalov's Theorem by means of Exercise 4.12.
The question states, for $\mathfrak{sl}_3(\mathbb{C})$ let $\Phi^+ = \{ \alpha, \beta, \gamma = \alpha + \beta \}$ with standard basis in terms of $\mathfrak{sl}_2$ triples. Let $\lambda = a \omega_\alpha + b \omega_\beta$ for the $\omega$s the fundamental weights. Then I have shown $\langle \lambda + \rho, \rho^\vee \rangle = 1 \iff a + b = -1$, where $\rho$ denotes half the sum of the positive roots, $\rho = \alpha + \beta = \gamma$.
Set $r, s \in \mathbb{C}$ not both zero and $u = ry_\alpha y_\beta + s y_\gamma \in \mathcal{U}(\mathfrak{n^-})$, the enveloping algebra for $\mathfrak{n}^- = \bigoplus_{\alpha \in \Phi^-} \mathfrak{g}_{\alpha}$. If $v^+$ is a $\lambda$ weight vector, suppose that $u \cdot v^+$ is a $\lambda - \rho$ weight vector. Show that $r(a+1) - s = 0$.
I believe I want to act on the vector with weight $\lambda - \rho = \lambda - (\alpha + \beta)$ to re-obtain a $\lambda$ weight vector and then the equality shall follow by comparison with with the description of $\lambda$ in terms of the fundamental weights. Is this approach correct?
I feel I am missing something, any suggestions would be appreciated.