Closed orbits for actions of maximal tori on homogeneous spaces

Abstract

Let G be a real algebraic group defined over Q, let be an arithmetic subgroup, and let T be any torus containing a maximal R-split torus. We prove that the closed orbits for the action of T on G/ admit a simple algebraic description. In particular, we show that if G is reductive, an orbit T x is closed if and only if x −1 T x is a product of a compact torus and a torus defined over Q, and it is divergent if and only if the maximal R-split subtorus of x −1 T x is defined over Q and Q-split. Our analysis also yields the following:• there is a compact K ⊂ G/ which intersects every T -orbit;• if rank Q G < rank R G, there are no divergent orbits for T .

On Topologically Big Divergent Trajectories

“…The result we present here generalizes [28]. A crucial step in [28] (and a result interesting in its own right) was showing that there is a fixed compact subset of G/Γ which intersects every diagonal orbit.…”

Divergent trajectories in arithmetic homogeneous spaces of rational rank two

“…The following useful criterion is similar to the one proved in [TW,Proposition 3.5] and can be deduced from its proof. PROPOSITION 4.2.…”

“…PROPOSITION 4.4. [TW,Proposition 3.3] There exists a neighborhood W 0 of zero in g such that for any g ∈ G, the span of Ad(g)g Z ∩ W 0 is horospherical. PROPOSITION 4.5.…”

“…What is the difference between these two cases? It seems that the heart of the matter is the rational rank of G. In [W1] it was conjectured by Weiss that if A is a diagonal subgroup, then the existence of divergent trajectories and of 'non-obvious' divergent trajectories depends only on the relation between the dimension of A and the rational rank of G. See Definitions 1.2 and 1.3 for the definitions of obvious and degenerate divergent trajectories, respectively, and [TW,W1,W2] for results regarding the conjecture.…”

“…We start by looking at a divergent trajectory under the action of a closed cone. In §4 a compactness criterion that can be deduced from [TW,§3] is stated. Using the compactness criterion we can attach to each element in the cone its 'reason for divergence'.…”