Abstract
We consider aspects of the relationship between nilpotent orbits in a semisim-ple real Lie algebra g and those in its complexification gi. In particular, we prove that two distinct real nilpotent orbits lying in the same complex orbit are incomparable in the closure order. Secondly, we characterize those g having non-empty intersections with all nilpotent orbits in gi. Finally, for g quasi-split, we characterize those complex nilpotent orbits containing real ones.
Highlights
1.1 Background and statement of resultsReal and complex nilpotent orbits have received considerable attention in the literature
There has been some interest in concrete descriptions of the poset structure on real nilpotent orbits in specific cases [2,3]
Complex nilpotent orbits are studied in algebraic geometry [4,5,6] and representation theory — in particular, Springer Theory [7]
Summary
Real and complex nilpotent orbits have received considerable attention in the literature. The former have been studied in a variety of contexts, including differential geometry, symplectic geometry, and Hodge theory [1]. (i) Every complex nilpotent orbit is realizable as the complexification of a real nilpotent orbit if and only if g is quasi-split and has no simple summand of the form so (2n+1, 2n −1). (ii) If g is quasi-split, a complex nilpotent orbit Q ⊆ g is realizable as the complexification of a real nilpotent orbit if and only if Q is invariant under conjugation with respect to the real form g ⊆ g. (iii) If 1, 2 ⊆ g are real nilpotent orbits satisfying ( 1) =( 2) , either 1= 2 or these two orbits are incomparable in the closure order
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
More From: Journal of Generalized Lie Theory and Applications
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.