Automorphisms of Multiplicity Free Hamiltonian Manifolds

Consider a connected compact Lie group K acting on a connected Hamiltonian manifold M . A measure for the complexity of M is half the dimension of the symplectic reductions of M , and it is natural to study Hamiltonian manifolds with low complexity first, starting with the case of complexity zero, the so-called multiplicity free manifolds (see [GS] or [MiFo]). It has been a longstanding problem to classify multiplicity free manifolds, and it is the purpose of this paper to complete this project. More specifically, Delzant conjectured in 1989 that any compact multiplicity free space is uniquely determined by two invariants: its momentum polytope P and its principal isotropy group L0. Evidence for this conjecture was Delzant’s celebrated classification of multiplicity free torus actions [De1], as well as further particular cases settled by Iglésias (K = SO(3), [Igl]), Delzant (rkK = 2, [De2]), and Woodward (transversal actions, [Wo1]). The main objective of this paper is, building upon work of Losev [Los], to complete the proof of Delzant’s conjecture (see Theorem 10.2). Once we know that a multiplicity free manifold is characterized by the combinatorial data (P, L0) it is natural to ask which pairs actually arise this way. In section 11 we show that this can be reduced to a purely local problem on P. More precisely, P has to “look” locally like the weight monoid of a smooth affine spherical variety (see Theorem 11.2). Since the latter class of varieties has been previously classified by Van Steirteghem and the author [KVS] this finishes the classification of multiplicity free manifolds. The proof of the Delzant Conjecture proceeds in two separate steps: first a local statement (ultimately due to Losev [Los]) and then a local-to-global argument (addressed in this paper). First, we describe briefly the local problem. Let t ⊆ k be a Cartan subalgebra. Then it is well known that the orbit space k∗/K can be identified with a Weyl chamber t ⊆ t. Thus the moment map m : M → k∗ gives rise to the invariant moment map ψ : M → t. By a celebrated theorem of Kirwan [Kir1], the image P = ψ(M) is a convex polytope for M compact. The local statement asserts now that two compact multiplicity free manifolds with the same momentum polytope P and the same principal isotropy group are isomorphic locally over P (see Theorem 2.4). Using techniques from [Sja], one can reduce this local problem to a statement about