We use geometric information obtained from the generic initial ideal of a hyperplane section of a surface in ${\bb P}^4$ not of general type to bound the degree of such a surface.

The main goal of the paper is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal Grassmannians of maximal isotropic subbundles as well as some globalizations of them. The used geometric tools overlap appropriate desingularizations of such Schubert subschemes and Gysin maps for such Grassmannian bundles. The main algebraic tools are provided by the families of $\tilde{Q}$- and $\tilde{P}$-polynomials introduced and investigated in the present paper. The key technical result of the paper is the computation of the class of the (relative) diagonal in isotropic Grassmannian bundles based on the orthogonality property of $\tilde{Q}$- and $\tilde{P}$-polynomials. Some relationships with quaternionic Schubert varieties and Schubert polynomials for classical groups are also discussed.

Let $G$ be either a split ${\rm SO}(2n+2)$, or a split adjoint group of type $E_n, (n=6,7,8)$, over a $p$-adic field. In this article we study correspondences arising by restricting the minimal representation of $G$ to various dual pairs in $G$.