[Paper Review] Hermitian flag manifolds and orbits of the Euclidean group

Philip Arathoon, James Montaldi|MIMS EPrints (University of Southampton)|Apr 25, 2018

Advanced Algebra and Geometry3 references3 citations

TL;DR

This paper establishes a geometric bijection between adjoint and coadjoint orbits of Lie groups of Euclidean type—semidirect products of a compact group H and a representation V—demonstrating that corresponding orbits are homotopy equivalent and, for the Euclidean group, even fiber bundle structures. It introduces Hermitian flag manifolds as geometric models for these orbits, providing a flag-based description of adjoint and coadjoint orbits of the Euclidean and orthogonal groups with complex structures on quotient spaces.

ABSTRACT

We study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic, we show that there is a geometrically defined bijection between the sets of adjoint and coadjoint orbits of such groups. In addition, we show that the corresponding orbits, although different, are homotopy equivalent. We also provide a geometric description of the adjoint and coadjoint orbits of the Euclidean and orthogonal groups as a special class of flag manifold which we call a Hermitian flag manifold. These manifolds consist of flags endowed with complex structures equipped to the quotient spaces that define the flag.

Motivation & Objective

• To establish a geometric bijection between adjoint and coadjoint orbits for groups of Euclidean type, where the adjoint and coadjoint representations are not isomorphic.
• To provide a geometric description of adjoint and coadjoint orbits of the Euclidean and orthogonal groups as Hermitian flag manifolds, incorporating complex structures on quotient spaces.
• To show that corresponding orbits under the bijection are homotopy equivalent, and for the Euclidean group, that one orbit is a vector or affine bundle over the other.
• To demonstrate that generic coadjoint orbits fiber over the symplectic manifold of oriented lines in R^n, and that these fibrations are symplectic with isotropic fibres.

Proposed method

• Define groups of Euclidean type as semidirect products G = H ⋉ V, where H is compact and V is a representation of H.
• Use the adjoint and coadjoint actions of G on its Lie algebra and dual to classify orbits via flag-type structures with complex or oriented quotient spaces.
• Construct a geometric bijection between adjoint and coadjoint orbits by analyzing stabilizers and orbit spaces, showing they are in one-to-one correspondence despite non-isomorphic representations.
• Prove homotopy equivalence of bijected orbits by showing they are fiber bundles with contractible fibres, using the structure of kernels of linear maps.
• Apply results from symplectic geometry to show that the orbit of oriented lines in R^n (D_n) is a fundamental symplectic manifold over which other orbits fiber.
• Use the flag bundle structure to define invariant symplectic forms on Hermitian flag manifolds, showing fibrations are symplectic or have isotropic fibres.

Experimental results

Research questions

• RQ1 Is there a natural geometric correspondence between adjoint and coadjoint orbits for non-semisimple Lie groups like the Euclidean group?
• RQ2 Can adjoint and coadjoint orbits of the Euclidean group be described as flag manifolds with additional geometric structures?
• RQ3 Do corresponding adjoint and coadjoint orbits have the same homotopy type, even when the representations are not isomorphic?
• RQ4 Can the symplectic geometry of these orbits be understood via fibrations over the standard symplectic manifold D_n of oriented lines in R^n?
• RQ5 What is the role of complex structures and orientations in defining the geometry of these orbit manifolds?

Key findings

• A geometric bijection exists between adjoint and coadjoint orbits of groups of Euclidean type, even though the adjoint and coadjoint representations are not isomorphic.
• Corresponding orbits under this bijection are homotopy equivalent, as they are fiber bundles with contractible fibres.
• For the Euclidean group, each coadjoint orbit is a vector or affine bundle over the corresponding adjoint orbit, and vice versa.
• The generic coadjoint orbits of the Euclidean group fiber over D_n, the symplectic manifold of oriented lines in R^n, with symplectic fibres isomorphic to Hermitian flag manifolds.
• The flag manifolds AffF(1̃; d₀−1, d₁^C, ..., d_k^C) and F(d₀−1, d₁^C, ..., d_k^C) admit invariant symplectic forms, and the fibrations are symplectic or have isotropic fibres.
• All adjoint and coadjoint orbits of groups of Euclidean type are submanifolds of the flag manifolds constructed for E(n), with SE(n) orbits being connected components of E(n) orbits.