Projective representations of real semisimple Lie groups and the gradient map

Let G be a real noncompact semisimple connected Lie group and let be a faithful irreducible representation on a finite-dimensional vector space V over . We suppose that there exists a scalar product on V such that , where and . Here, denotes the Lie algebra of G, denotes the connected component of the orthogonal group containing the identity element and denotes the set of symmetric endomorphisms of V with trace zero. In this paper, we study the projective representation of G on arising from . There is a corresponding G-gradient map . Using G-gradient map techniques, we prove that the unique compact G orbit inside the unique compact orbit in , where U is the semisimple connected compact Lie group with Lie algebra , is the set of fixed points of an anti-holomorphic involutive isometry of and so a totally geodesic Lagrangian submanifold of . Moreover, is contained in . The restriction of the function , where is an -invariant scalar product on , to achieves the maximum on the unique compact orbit of a suitable parabolic subgroup and this orbit is connected. We also describe the irreducible representations of parabolic subgroups of G in terms of the facial structure of the convex body given by the convex envelope of the image