We apply the theory of windows, as developed by Halpern-Leistner and by Ballard, Favero and Katzarkov, to study certain moduli spaces and their derived categories. Using quantization and other techniques we show that stable GIT quotients of $(\mathbb{P}^1)^n$ by $PGL_2$ over an algebraically closed field of characteristic zero satisfy a rare property called Bott vanishing, which states that $\Omega^j_Y \otimes L$ has no higher cohomology for every j and every ample line bundle L. Similar techniques are used to reprove the well known fact that toric varieties satisfy Bott vanishing. We also use windows to explore derived categories of moduli spaces of rank-two vector bundles on a curve. By applying these methods to Thaddeus' moduli spaces, we find a four-term sequence of semi-orthogonal blocks in the derived category of the moduli space of rank-two vector bundles on a curve of genus at least 3 and determinant of odd degree, a result in the direction of the Narasimhan conjecture.