### Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, by Ivan Panin and Charles Walter

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^{n} = HGr(1,n+1). For a symplectically
oriented cohomology theory A, including oriented theories but also hermitian
K-theory, Witt groups and symplectic and special linear algebraic cobordism, we
have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Pontryagin classes for
symplectic bundles. They satisfy a 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.

Ivan Panin <paniniv@gmail.com>

Charles Walter <Charles.Walter@unice.fr>