On the orbit space of a compact linear Lie group with commutative connected component

Abstract:

This paper is devoted to the study of topological quotients of compact linear Lie groups. More precisely, it investigates the question of when such a quotient is a topological or a smooth manifold.

The topological quotient of a finite linear group was studied by Mikhaĭlova in 1984. Here the connected component $G^0$ of the original Lie group $G$ is assumed to be a torus of positive dimension.

The main method used here is to consider the weight system corresponding to the decomposition of the representation of the commutative group $G^0$ into irreducible representations. In §8 an arbitrary linear group is reduced to a linear group of special type, namely, one with a $2$-stable weight system (for the definition and properties of $q$-stable sets of vectors, where $q\in \mathbb {N}$, see §§1, 4). The main results for a group with a $2$-stable weight system are stated in the Introduction (Theorems 1.3–1.8) and proved in §§6 and 7.