Torus orbits in G/P

Let G be a complex semisimple Lie group of rank l, with fixed Borel subgroup B and maximal torus H. Let P be a standard parabolic subgroup. The torus H acts on G/P by gP → hgP. The closure X in G/P of an orbit {hgP\h ∈ H} is called a torus orbit if it is l-dimensional and satisfies a certain genericity condition; it is a rational algebraic variety whose structure is intimately related to Lie theory, symplectic geometry, and the theory of convex bodies. This paper presents: (1) an abstract description of the torus orbit X by means of a rational polyhedral fan; (2) a description of the torusinvariant divisor whose linear system provides a natural embedding (the Plucker embedding) of X into a projective space; (3) a discussion of the correspondence between this divisor and the momentum mapping associated to the action on X of the compact torus T ⊂ H (4) a list of generators of the ideal defining the Plucker embedding; (5) a formula for the intersection multiplicity of certain important torus invariant divisors on X.