  A smooth projective variety $Y$ is said to satisfy Bott vanishing if
$\Omega_Y^j\otimes L$ has no higher cohomology for every $j$ and every ample
line bundle $L$. Few examples are known to satisfy this property. Among them
are toric varieties, as well as the quintic del Pezzo surface, recently shown
by Totaro. Here we present a new class of varieties satisfying Bott vanishing,
namely stable GIT quotients of $(\mathbb{P}^1)^n$ by the action of $PGL_2$,
over an algebraically closed field of characteristic zero. For this, we use the
work done by Halpern-Leistner on the derived category of a GIT quotient, and
his version of the quantization theorem. We also see that, using similar
techniques, we can recover Bott vanishing for the toric case.
