Quaternionic Grassmannians and Borel classes in algebraic geometry

The quaternionic Grassmannian $\mathrm{HGr}(r,n)$ is the affine open subscheme of the usual Grassmannian parametrizing those $2r$-dimensional subspaces of a $2n$-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have $\mathrm{HP}^{n} = \mathrm{HGr}(1,n+1)$. For a symplectically oriented cohomology theory $A$, including oriented theories but also the Hermitian $\mathrm{K}$-theory, Witt groups, and algebraic symplectic cobordism, we have $A(\mathrm{HP}^{n}) = A(\operatorname{pt})[p]/(p^{n+1})$. Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank $2$ symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes.
The cell structure of the $\mathrm{HGr}(r,n)$ exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is $\mathrm{HP}^{n}$ where the cell of codimension $2i$ is a quasi-affine quotient of $\mathbb{A}^{4n-2i+1}$ by a nonlinear action of $\mathbb{G}_{a}$.