Field extensions and isotropic subspaces in symplectic geometry (Q917906)

Let \(L/k\) be a finite field extension, \((V,B)\) a symplectic space over \(L\), and \(Sp_ L(V)\) the corresponding symplectic group. For any nonzero \(k\)- linear functional \(\phi: L\to k\) consider the \(k\)-valued symplectic form \(B'=\phi \circ B\) on V. \(Sp_ L(V)\) acts on \(B'\)-isotropic \(k\)-subspaces in \(V\). This paper determines the number of orbits of \(d\)-dimensional such subspaces for \(d\leq \dim_ kV\). This number turns out to be finite if \([L:k]=2\), finite only for \(d=0,1\) or \(1/2 \dim_ kV\) if \([L:k]=3\), whereas for \([L:k]\geq 4\) it is always infinite if the fields are infinite. In the finite cases, the precise structure of the orbits is determined.