Abstract

We consider aspects of the relationship between nilpotent orbits in a semisim-ple real Lie algebra g and those in its complexification giƒ. 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ƒ. 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]

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Summary

Background and statement of results

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

Structure of the article
Nilpotent orbits
The closure orders
Partitions of nilpotent orbits
The complexification map
Surjectivity
Fibres
Full Text
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