Classification of invariant valuations on the quaternionic plane

Abstract

We describe the orbit space of the action of the group $\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds ${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than $2$. We then give a complete classification of valuations on the quaternionic plane ${\mathbb{H}}^2$ which are invariant under the action of the group $\mathrm{Sp}(2)\mathrm{Sp}(1)$.

26 pages; minor changes

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