On torus quotients of Schubert varieties in orthogonal Grassmannian-II

Abstract: Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $\alpha_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.