On subgroups normalized by EO(2l, R)

It is shown that the problem of describing those subgroups in the general linear group GL( n, R ) which are normalized by a classical group is much more difficult than believed previously. For the case of even orthogonal groups, a thorough level calculation is performed, which shows that, even under the assumption 2 ∈ R *, the level of a subgroup H ≤ GL(2 l, R ), l ≥ 3, normalized by EO(2 l, R ), is determined by three ideals ( A, B, C ) in R rather than by two ideals, as was generally believed. These ideals are related by C 2 ≤ A = B ∩ C , and triples of such ideals are said to be admissible. Here, A is the level of H with respect to the linear transvections t ij (ξ), and B is the level of H with respect to the orthogonal transvections T ij (ξ). The definition of the third level component is a little more complicated. In an appropriate realization, the Lie algebra of the even orthogonal group consists of matrices antisymmetric with respect to the skew diagonal. The component C is the level of H with respect to the complementary invariant subspace, which consists of matrices symmetric with respect to the skew diagonal. With any admissible triple ( A, B, C ) we associate a relative elementary subgroup EEO(2 l, R, A, B, C ), which is normalized by EO(2 l, R ) and, moreover, is EO(2 l, R )-perfect.