Invariant Monge–Ampère equations on contactified para–Kähler manifolds

Abstract We develop a method for describing invariant PDEs of Monge–Ampère type in the sense of Lychagin and Morimoto (MAE) on a homogeneous contact manifold N of a semisimple Lie group G , which is the contactification of the homogeneous symplectic manifold $$M = G/H = \textrm{Ad}_G Z \subset \mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mtext>Ad</mml:mtext> <mml:mi>G</mml:mi> </mml:msub> <mml:mi>Z</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> , where M is the adjoint orbit of a splittable closed element Z of the Lie algebra $$\mathfrak {g}= {{\,\textrm{Lie}\,}}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mspace/> <mml:mtext>Lie</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The method is then applied to a ten-dimensional semisimple orbit M of the exceptional Lie group $$\textsf{G}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and a complete list of mutually non-equivalent MAEs on N is obtained.