Seminar talk, 19 October 2016

From Geometry of Differential Equations
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Speaker: Dmitri Alekseevsky

Title: Second order partial differential equations with a simple symmetry group

Let [math]\displaystyle{ (M = G/H, C) }[/math] be the homogeneous space of a simple Lie group with an invariant contact distribution, which locally defined by a contact form [math]\displaystyle{ \theta }[/math]. The non-degenerate 2-form [math]\displaystyle{ \omega := d\theta |_{C} }[/math] is defined up to a conformal multiplier. The manifold [math]\displaystyle{ M^{(1)} }[/math] of Lagrangian planes in [math]\displaystyle{ C_p , \, p \in M }[/math] is the total space of the bundle [math]\displaystyle{ \pi : M^{(1)} \to M }[/math]. The fiber is the Grassmannian [math]\displaystyle{ LGr_n }[/math] of the Lagrangian planes of symplectic vector space [math]\displaystyle{ \mathbb{R}^{2n} \simeq C_p }[/math]. The group [math]\displaystyle{ G }[/math] acts on [math]\displaystyle{ M^{(1)} }[/math] as the automorphisms group, and the [math]\displaystyle{ G }[/math]-invariant hypersurfaces [math]\displaystyle{ E \subset M^{(1)} }[/math] are 2nd order equations with the symmetry group [math]\displaystyle{ G }[/math].

The talk will discuss the description of such hypersurfaces in the case when [math]\displaystyle{ M=G/H }[/math] is the adjoint manifold of a simple complex Lie group [math]\displaystyle{ G }[/math], i.e., the orbit [math]\displaystyle{ M = \mathrm{Ad}_G [E_{\mu}] }[/math] of a highest vector of Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] in the projective space [math]\displaystyle{ P\mathfrak{g} }[/math].

The talk is based on the joint work with Jan Gutt, Gianni Manno, and Giovanni Moreno.