Quaternionic Grassmannians and Pontryagin Classes in Algebraic Geometry

The quaternionic Grassmannian HGr(r, n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP = HGr(1, n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and algebraic symplectic cobordism, we have A(HP) = A(pt)[p]/(p). We define Pontryagin classes for symplectic bundles. They satisfy the splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Pontryagin classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Pontryagin classes. The cell structure of the HGr(r, n) exists in the cohomology, but it is difficult to see more than part of it geometrically. The exception is HP where the cell of codimension 2i is a quasi-affine quotient of A by a nonlinear action of Ga.