• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.671-683
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• 2022

The aim of this paper is to deal with the realization problem of a given Lagrangian submanifold of a symplectic manifold as the fixed point set of an anti-symplectic involution. To be more precise, let (X, ω, µ) be a toric Hamiltonian T-space, and let ∆ = µ(X) denote the moment polytope. Let τ be an anti-symplectic involution of X such that τ maps the fibers of µ to (possibly different) fibers of µ, and let p₀ be a point in the interior of ∆. If the toric fiber µ^-1(p₀) is real Lagrangian with respect to τ, then we show that p₀ should be the origin and, furthermore, ∆ should be centrally symmetric.