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 $LGr_{n}$ 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 $M=\mathrm {Ad} _{G}[E_{\mu }]$ of a highest weight 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