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If $G=Gr(k,n)$ denotes the Grassmannian of k-dimensional subspaces in $V= \mathbb C^n$, representation theory gives us a Bott-type result for the cohomology groups $H^q(G, \Omega^p(k))$ of the twisted differentials.\ In particular, as quoted in this question here on MO, for $G(2,n)$, Lemma 0.1 of this paper (http://arxiv.org/pdf/alg-geom/9306010v2.pdf) gives a complete answer on this topic.

I was now wondering if similar results exists for the weighted Grassmannian $wGr(2,5)$ introduced by Corti-Reid here. In particular I am interested in the vanishing of $H^i(wG, \Omega^2(k))$, for $k<0$, $i=1,2,3$.

I suspect this being true (just to say, the corresponding cohomology groups on the 'straight' Grassmannian vanishes and, naively speaking, having non-trivial weights gives 'more room' for vanishing); anyway, I haven't being able neither to find a good reference!

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