Let be a standard subspace in the complex Hilbert space , and let be a unitary representation of a finite-dimensional Lie group. We assume the existence of an element such that is the modular group of and that the modular involution normalizes . We want to determine the semigroup . In previous work, we have seen that its infinitesimal generators span a Lie algebra on which defines a 3-grading, and here we completely determine the semigroup under the assumption that defines a 3-grading on . Concretely, we show that the -eigenspaces contain closed convex cones , such that
where is the stabilizer of . To obtain this result, we compare several subsemigroups of G specified by the grading and the positive cone of U. In particular, we show that the orbit with the inclusion order is an ordered symmetric space covering the adjoint orbit , endowed with the partial order defined by .
"Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces." Kyoto J. Math. 62 (3) 577 - 613, September 2022. https://doi.org/10.1215/21562261-2022-0017