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 torus-invariant divisor whose linear system provides a natural
embedding (the Plücker 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 Plücker embedding; (5) a formula for the
intersection multiplicity of certain important torus invariant divisors on
X.
