Action of classical groups on the varieties associated with skew field extensions

Abstract

Let $L|K$ be a finite skew field extension, K commutative and V a finite dimensional vector space over L. We study the action of general L-linear groups on the set of K-subspaces of V and the action of unitary groups on the set of $f'$-isotropic K-subspaces of V. In the latter case let $(V,f)$ be a vector space endowed with a regular $\varepsilon$-hermitian form, $φ:L→K$ a K-linear map and $f':=φ○ f$. We show that for linear groups the number of orbits depends only on the degree of the field extension and on the dimension of V. The orbits are classified completely when $[L:K]≤ 3$. For unitary groups in general the number of orbits depends also on the underlying fields and on the map φ. We discuss in more detail the quadratic case for some particular fields.