Witt groups of isotropic Grassmann bundles

The goal of this work is to compute the Witt groups of maximal isotropic Grassmann bundles, that is, schemes parametrizing subbundles of maximal rank of a fixed vector bundle which are isotropic with respect to a given symmetric or symplectic bilinear form. The case of ordinary Grassmannians has been accomplished by Balmer and Calmès by investigating the boundary map in the localization long exact sequence of Witt groups. We prove that the total Witt group is parametrized by even shifted Young diagrams in the orthogonal case and by almost even shifted Young diagrams in the symplectic case.