# Seminar talk, 19 October 2016

Speaker: Dmitri Alekseevsky

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

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

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

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

References:
arXiv:1606.02633