Seminar talk, 19 October 2016

Speaker: Dmitri Alekseevsky

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

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

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

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

References:
arXiv:1606.02633