One of the most mysterious aspects of Saito’s theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well as the resulting weight filtration. The main tool is a categorical \({\mathfrak {sl}}_2\) action on the category of \({\mathcal {D}}_h\)-modules on Grassmannians. Along the way we also clarify the interaction of kernels for \({\mathcal {D}}_h\)-modules with the associated graded functor. Both of these results may be of independent interest.

Saito的Hodge模块理论中最神秘的方面之一是在开放式嵌入下伴随Hodge模块前推的Hodge和权重过滤。在本文中，我们考虑由成对的横向子空间给出的互补Grassmannian乘积中的开放嵌入。从几何表示理论和同源结不变式的角度来看，在这种开放式嵌入下结构捆的前推是一个重要的霍奇模块。我们针对诱导的霍奇过滤以及由此产生的重量过滤计算了这种前推的相关等级。主要工具是对\（{\ mathcal {D}} _ h \）类别的分类\（{\ mathfrak {sl}} _ 2 \）操作-在Grassmannians上的模块。在此过程中，我们还阐明了\（{\ mathcal {D}} _ h \）-模块与相关的分级函子的内核之间的交互作用。这两个结果都可能具有独立的意义。