Action on flag varieties: 2-dimensional case (Q1208690)

Let \(A\) be a finite dimensional commutative semisimple algebra over a field \(k\), and let \(V\) be a finitely generated \(A\)-module. We examine the action of the general linear group \(GL_ A(V)\) on the set of flags of \(k\)-subspaces of \(V\). Also, let \((V,B)\) be a finitely generated `symplectic module' over \(A\). We also investigate the action of the symplectic group \(Sp_ A(V,B)\) on the set of flags of \(B'\)-isotropic \(k\)-subspaces of \(V\), where \(B'=\varphi \circ B\) is the \(k\)-symplectic form induced by a nonzero \(k\)-linear map \(\varphi:A\to k\). Studied earlier were the cases of \(GL_ A(V)\) acting on the set of \(k\)-subspaces of \(V\) and of \(Sp_ A(V,B)\) acting on the set of \(B'\)-isotropic \(k\)- subspaces of \(V\). So this paper is a natural extension of the previous works. In both cases, the orbits are completely classified in terms of certain integer invariants provided that \(\dim_ k A=2\), from which one can determine the precise structure of orbits, compute the exact number of orbits and give typical representatives for each orbit, etc.