Elliptic domains in Lie groups

An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic elements appear naturally in the geometry of causal symmetric spaces and in representation theory. We characterize stably elliptic elements in terms of the fixed point algebra of $\mathrm{Ad}(g)$ and show that the connected components of the set $G^{se}$ of stably elliptic elements can be described in terms of the Weyl group action on a compactly embedded Cartan subalgebra. In the case of simple hermitian Lie groups we relate stably elliptic elements to maximal invariant cones and the associated subsemigroups. In particular we show that the basic connected component $G^{se}(0)$ can be characterized in terms of the compactness of order intervals and that $G^{se}(0)$ is globally hyperbolic with respect to the induced biinvariant causal structure.