I want to compute the composition factor multiplicities of the Verma module $M(0)$ for the complex Lie algebra $\mathfrak{sl}_3$ using elementary methods. I already have some results: $[M(w. 0) : L(s_1 s_2 s_1. 0)] = 1$ for all $w$ in the Weyl group. Using the Jantzen sum formula, I have found that $[M(0) : L(\lambda)] = 1$ for $\lambda = 0,\ s_1. 0,\ s_2. 0, \ s_1 s_2 s_1.0$. However, for $\lambda = s_1 s_2. 0,\ s_2 s_1. 0$, I only obtain $[M(0) : L(\lambda)] \leq 2$. How can I show that these multiplicities are exactly 1?
I would appreciate any elementary approach or insight to resolve this.
