On the cohomology of the Lie algebra of formal vector fields preserving a flag (Q796149)

Let \({\mathcal A}_{n,r}=\{\sum^{r}_{i=1}f_ i(x_ 1,...,x_ r) \partial /\partial x_ i+\sum^{n}_{i=r+1}f_ i(x_ 1,...,x_ n) \partial /\partial x_ i| f_ i\) is a formal power series in its variable\(s\}\) and denote \({\mathcal A}_ r={\mathcal A}_{r,r}\). The author shows that the Lie algebra \({\mathcal A}_{n,r}\) is r-connected, i.e. \(H^ i({\mathcal A}_{n,r},{\mathbb{R}})=0\) for \(0<i\leq r\). Moreover, it is shown that there is a canonical isomorphism of \(H^ i({\mathcal A}_{n,r},{\mathbb{R}})\) with \(H^ i({\mathcal A}_ r,{\mathbb{R}})\) for \(i\leq n-r\). It is still an open question if this is true for \(i\leq 2n\).