On the invariant theory and geometry of compact linear groups of cohomogeneity⩽3

Copolarity of isometric actions

“…This follows from results in Section 5 in [Str94] or from the computation in Proposition 7.12 in [GTa]. (These representations are listed in the table of Theorem 1.1; for the classification of cohomogeneity 3 representations, see [Yas86] for the irreducible case and [Uch80] for the reducible case; the nonpolar cohomogeneity 3 representations are listed in Table II in [Str94] and are essentially proved to have copolarity 1 in Theorem 5.1 of the same paper.)…”

“…Notice that Theorems 1.1 and 1.3 cease to hold if the representation is not irreducible, as can be seen by taking the 7-dimensional representation of U(2) given by the direct sum of the vector representation on C 2 and the adjoint representation on su(2). In fact this representation has copolarity one (see [Str94]) and can easily be seen not to be taut by the methods of [GTa], [GT03]. (It is interesting to remark that this representation still has cohomogeneity 3.)…”

“…It is interesting to notice that sometimes the group G can be enlarged to another groupĜ that has the same orbits in M as G but has a larger principal isotropy subgroupĤ, and then theĤ-fixed point set is contained in the H-fixed point set (see [Str94]). On the other hand, the concept of copolarity is purely geometrical, and so it coincides for orbit equivalent actions.…”

“…In this paper we examine this problem in the case of orthogonal representations. Examples of representations of nontrivial copolarity and minimal k-sections appear naturally in the framework of the reduction principle in compact transformation groups (see [GS00], [SS95], [Str94] for that principle). In fact, in [GT02] the reduction principle was used to describe the geometry of the irreducible representations in the table of Theorem 1.1 below, which have copolarity 1.…”

A slice theorem for singular Riemannian foliations, with applications

“…If Γ is a (say finite) group containing such non-liftable isometries, then one may define the leaves of F to be the inverse images of Γ-orbits under the quotient map V → V /F 0 . Some concrete homogeneous examples of such F 0 may be extracted from the classification of orbit spaces of cohomogeneity 3 representations, see [Str94,Lemma 6.1]. They include polar representations with generalized Weyl group of the form Z/2 × D 4 or Z/2 × D 6 , where D k denotes the dihedral group with 2k elements.…”