Continuous selection of Lagrangian subspaces (Q6622094)

Polarizing subalgebras of Lie algebras and Lagrangian subspaces of presymplectic vector spaces provide a nice illustration of the applications of linear algebra to areas such as symplectic geometry, geometric quantization, and in particular explicit constructions of unitary representations associated to coadjoint orbits of Lie groups. These explicit constructions are important in representation theory, integrable systems, linear partial differential equations, and construction of frames associated to Lie group representations.\N\NThe authors study continuous selections of the set-valued map that takes every skew-symmetric bilinear form on a vector space to its corresponding set of maximal isotropic subspaces. Applications are made to establishing continuity properties of the Vergne polarizing subalgebras of completely solvable Lie algebras in terms of Schubert cells of suitable Grassmann manifolds.