Symplectic reduction and Lagrangian submanifolds of Gr(1, n )

New examples of Lagrangian submanifolds of the complex Grassmannian Gr(1, n) with the standard Ka<spacing diaeresis>hler form are presented. The scheme of their construction is based on two facts: first, we put forward a natural correspondence between the Lagrangian submanifolds of a symplectic manifold obtained by symplectic reduction and the Lagrangian sub- manifolds of a large symplectic manifold carrying a Hamiltonian action of some group, to which this reduction is applied; second, we show that for some choice of generators of the action of T k on Gr(1, n), k = 2 , ... , n 1 , and for suitable values of the moment map there exists an isomorphism Gr(1, n)//Tk similar to= tot(P(tau) x <middle dot> <middle dot> <middle dot> x P(tau)-> Gr(1, n k)), where the total space of the Cartesian product of k copies of the projectivization of the tautological bundle tau -> Gr(1, n k) is on the right. Combining these two facts we obtain a lower bound for the number of topologically distinct smooth Lagrangian submanifolds in the original Grassmannian Gr(1, n). Bibliography: 5 titles.