Note on semisymplectic actions of Lie groups

A semisymplectic action of a Lie groups on a symplectic manifold is one whe re each element of the group acts either symplectically or antisymplectical ly. We find conditions that a semisymplectic action descends to an action o n the symplectic reduced spaces. We consider a few examples, and in particu lar apply these ideas to reduction of N-body systems with Galilean invarian ce. (C) 2000 Academie des sciences/Editions scientifiques et medicales Else vier SAS.