A Morse Theoretic Proof of Poisson Lie Convexity

Abstract

Let K be a connected compact Lie group that acts on the connected compact symplectic manifold (X, ω) preserving the symplectic form. If for every ξ in the Lie algebra p of K, the vector field ξx(x) = d/dt| _t=o exp(tξ) · x is Hamiltonian relative to a function, say Hξ, then the map Φ : X → p* (to the dual p* of p) defined by

$$H_\xi (x) = \left\langle {\xi ,\Phi (x)} \right\rangle $$

is called momentum mapping for the action of K. If the momentum mapping is equivariant relative to the given action of K on X and the coadjoint action of K on p*, the action is called Hamiltonian. A remarkable property of this map was discovered by Guillemin-Sternberg [GS1,GS2] and Kirwan [Ki2]. It asserts that if T is a maximal torus of K and t ^*₊ is a positive Weyl chamber, then Φ(X) ∩ t ^*₊ is a convex polytope. This theorem was first proved as follows. The image Φ(X) ∩ t ^*₊ was shown to be a finite union of compact convex polytopes in [GS1], and a convex polytope for X a Kähler manifold [GS2]. From the partial result in [GS1], Kirwan [Ki2]deduced convexity by appealing to her Morse theory (developed in [Ki1]) for||Φ||²