A Klein Cartan programme for differential equations and extrinsic geometries in flag manifolds

In 1872 Klein declared the Erlangen programme to understand various geometries
in a unified manner via transformation groups as homogeneous spaces, then in 1920's Cartan invented the notion of espace généralisé (principal bundle with Cartan connection in modern terminology)
to treat still group theoretically not only the homogeneous spaces but also inhomogeneous
spaces such as Riemannian geometries, conformal or projective differential geometries.
With modern approaches to general equivalence problems of geometric structures we have now
a general transparent view to intrinsic geometries.

In this talk we propose a Klein Cartan programme for differential equations in the framework of nilpotent
geometry and analysis.
In particular, we show a categorical correspondence between integrable overdetermined systems of linear partial differential equations and submanifolds in flag manifolds.
We then have a general method to find the invariants of a submanifold in a flag manifold,
based on an algebraic harmonic theory and the moving frame method, in the case when the relevant
Lie algebra is semi-simple.