Constructing projective varieties in weighted flag varieties II

We give the construction of weighted Lagranngiann Grassmannians$wLGr(3,6)$ and weighted partial $A_3$ flag variety $wF{l}_{1,3}$ coming from the symplectic Lie group $Sp(6,\mathbb{C})$ and the general linear group $GL(4,\mathbb{C})$ respectively. We give general formulas for their Hilbert series in terms of Lie theoretic data. We use them as key varieties (Format) to construct some families of polarized 3-folds in codimension 7 and 9. At the end, we list all the distinct weighted flag varieties in codimension $4\le c\le 10$.