The main result of this paper is a quasi-hamiltonian analogue of a special case of the O'Shea-Sjamaar convexity theorem for usual momentum maps. We denote by U a simply connected compact connected Lie group and we fix an involutive automorphism of maximal rank on this Lie group (such an automorphism always exists). We then denote by M a quasi-hamiltonian U-space and we prove that the image under the momentum map of the fixed-point set of a form-reversing compatible involution of M is a convex polytope, which is in fact equal to the full momentum polytope. This theorem was announced in arXiv:math/0609517v1. As an application, we obtain an example of lagrangian subspace in representation spaces of surface groups.
A symplectic approach to van den Ban's convexity theorem
设G是复半单李群，τ是与Cartan对合可交换的复反线性对合.若H表示G中τ-不动点的连通子群，且K是极大紧的，则G/K中的每个H-轨道都可以具有Evens和Lu所描述的Poisson结构.我们考虑辛叶的某些这样的H-轨道与一个自然的哈密顿环面行动。Hilgert-Neeb-Plank的一个辛凸性定理则导出了（复）半单对称空间的van den Ban凸性定理。
A convexity theorem for the real part of a Borel invariant subvariety
M. Brion证明了由Hamilton群作用的复化所保持的紧致积分Kaehler流形的不可约子簇的矩映射象的凸性结果。V. Guillemin和R. Sjamaar将这一结果推广到仅由博雷尔子群保持的不可约子簇。在另一个方向，L。O'Shea和R. Sjamaar证明了由反辛对合固定的子流形的矩映射象的凸性结果。类似于Guillemin和Sjamaar对Brion定理的推广，本文推广了O'Shea和Sjamaar的结果，证明了博雷尔子群所保持的不可约子簇的对合不动集的矩映射象的一个凸性定理.
Hamiltonian actions on 0-shifted cosymplectic groupoids
Instituto de Matemática e Estatística, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学数学与统计研究所）；Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计研究所）
