The action of the symplectic group associated with a quadratic extension of fields (Q1806096)

Let \(L\) be a field and \((V,f)\) a regular alternating space over \(L\). It is known that indecomposable subspaces, i.e., subspaces not splitting into the orthogonal sum of two proper subspaces, are either lines or hyperbolic planes. If \(K\) is a subfield of \(L\), then \(V\) is a vector space over \(K\). The authors ask the question, when is a \(K\)-subspace of \(V\) indecomposable? The classification of indecomposable \(K\)-subspaces allows the determination of the isometry classes in the set of \(K\)-subspaces of \(V\), which are orbits of the symplectic group \(\text{Sp}_L(V,f)\) in such a set. In this paper, the authors classify indecomposable \(K\)-spaces under the hypothesis that the extension \(L/K\) is quadratic. Since \(L/K\) is quadratic, every subspace \(W\) of \(V\) splits into the direct sum \(W= \text{comp}_L W\oplus W'\), where \(\text{comp}_LW\) denotes the largest \(L\)-subspace of contained in \(W\) and \(W'\) is a \(K\)-subspace of \(V\) (called a \(K\)-substructure) generated by vectors which are linearly independent over \(L\). Hence a pair \((m,n)\) of integers can be associated with \(W\), namely \(m= \dim_L \text{comp}_LW\) and \(n= \dim_KW'\). Thus \(\dim_KW= 2m+n\). \(K\)-subspaces of type \((m,0)\) and \((0,n)\) are \(L\)-subspaces and \(K\)-substructures, respectively. For a \(K\)-subspace \(W\) of type \((m,n)\) with \(m>0\), the authors prove that a necessary condition in order to which \(W\) is indecomposable is that \(m\leq 2\). Moreover, there is an indecomposable \(K\)-subspace of type \((1,n)\) just if \(n\not\equiv 0\pmod 4\) and an indecomposable \(K\)-subspace of type \((2,n)\) just if \(n\equiv 0\pmod 4\). Two indecomposable \(K\)-subspaces of the same type \((m,n)\), \(m=1,2\), are always isometric. Hence, in contrast with the case of \(K\)-substructures, the isometry classes in the set of indecomposable \(K\)-subspaces with nontrivial \(L\)-component do not depend on the field \(K\) and the number of isometry classes is precisely \(\dim_L V-1\).