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We study various kinds of Grassmannians or Lagrangian Grassmannians over$\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, all of which can be expressed as$\mathbb{G}/\mathbb{P}$ where $\mathbb{G}$ is a classical group and$\mathbb{P}$ is a parabolic subgroup of $\mathbb{G}$ with abelian unipotentradical. The same Grassmannians can also be realized as (classical) compactsymmetric spaces $G/K$. We give explicit generators and relations for the deRham cohomology rings of $\mathbb{G}/\mathbb{P}\cong G/K$. At the same time wedescribe certain filtered deformations of these rings, related to Cliffordalgebras and spin modules. While the cohomology rings are of our primaryinterest, the filtered setting of $K$-invariants in the Clifford algebraactually provides a more conceptual framework for the results we obtain.

Clifford algebras, symmetric spaces and cohomology rings of Grassmannians