Orthogonal linear group–subgroup pairs with the same invariants

The main theorem of Galois theory implies that there are no finite group–subgroup pairs with the same invariants. On the other hand, if we consider linear reductive groups instead of finite groups, the analogous statement is no longer true: There exist counterexample group–subgroup pairs with the same invariants. However, it is possible to classify all these counterexamples for certain types of groups. In [S. Solomon, Irreducible linear group–subgroup pairs with the same invariants, J. Lie Theory 15 (2005), 105–123], we provided the classification for connected complex irreducible groups, and, in this paper, for connected complex reductive orthogonal groups, i.e., groups that preserve some nondegenerate quadratic form.