I recently ran into a 30+ years old literature by Anderson and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Anderson-Jantzen] say that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by
$$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: 
$$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} 
V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i \\
0 & \text{otherwise}
\end{cases}
$$
Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then 
$$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$. 

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.