Let G be a connected Lie group
and 𝒜 an arbitrary, not necessarily connected Lie subgroup of Aut (G).
For a class of groups G described below, the authors examine the bounded
𝒜orbits in G (those with compact closure) and the group of bounded elements
B(G,𝒜) = {x ∈ G : 𝒜_{x} is bounded}. They first show that G may be split into closed,
𝒜invariant “layers” terminating at B(G,𝒜) whose properties insure that (i)
B(G,𝒜) is closed, (ii) every finite, 𝒜invariant Borel measure must have
supp μ ⊆ B(G,𝒜), and (iii) if x ∈ B(G,𝒜), there is such a measure μ with
x ∈ supp μ. Using these results, they prove a number of density theorems of the
following sort.
Theorem 1.1. Let ℬ⊆𝒜 be arbitrary, not necessarily connected Lie subgroups of
Aut (G) such that 𝒜∕ℬ has finite volume (or is compact). For any x ∈ G, ℬ_{x} is
bounded ⇔𝒜_{x} is bounded.
Theorem 1.2. Let B ⊆ A be arbitrary, not necessarily connected, closed subgroups of
G such that A∕B has finite volume (or is compact). Let α ∈ Aut (G) be arbitrary.
Then the displacement set disp(α,B) = {α(x)x^{−1} : x ∈ B} is bounded ⇔disp (α,A) is
bounded.
The authors prove these results for G whose Levi factor is faithfully
represented; there are indications that they remain true for all connected G. The
proofs devolve to questions about faithful linear representations of certain
nonconnected groups. Recent results of G. Hochschild show that G ×_{σ} Aut (G) is
faithfully represented if (i) the Levi factor of G is faithfully represented,
and (ii) the nilradical is simply connected. In the authors’ work it is crucial
to know that the representation can be chosen so that G × I (if nol all of
G ×_{σ} Aut (G)) is mapped to a closed subgroup of GL (V ); this is proved by
modifying Hochschild’s proofs, coupling them with methods developed by M.
Goto.
Furthermore, invariant finite Borel measures play a large role: Lie theory
yields information which drastically restricts the possible locations of such
measures. In crucial places the converse is true: invariant measure arguments
seem necessary to obtain algebraic and geometric information about the
actions.
