Strongly real adjoint orbits of complex symplectic Lie group

2024/11/14 by Lohan, Tejbir, Maity, Chandan
#15B30 #20E45 [2020] #FOS: Mathematics #Group Theory (math.GR) #Primary: 15A21 #Representation Theory (math.RT) #Secondary: 22E60 #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2411.09575

Abstract

We consider the adjoint action of the symplectic Lie group Sp(2n,ℂ) on its Lie algebra \mathfraksp(2n,ℂ). An element X ∈ \mathfraksp(2n,ℂ) is called Ad_Sp(2n,ℂ)-real if -X = Ad(g)X for some g ∈ Sp(2n,ℂ). Moreover, if -X = Ad(h)X for some involution h ∈ Sp(2n,ℂ), then X ∈ \mathfraksp(2n,ℂ) is called strongly Ad_Sp(2n,ℂ)-real. In this paper, we prove that for every element X ∈ \mathfraksp(2n,ℂ), there exists a skew-involution g ∈ Sp(2n,ℂ) such that -X =Ad(g)X. Furthermore, we classify the strongly Ad_Sp(2n,ℂ)-real elements in \mathfraksp(2n,ℂ). We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.

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