### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 251-292 |

Number of pages | 42 |

Journal | Pacific Journal of Mathematics |

Volume | 149 |

Issue number | 2 |

State | Published - 1991 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*149*(2), 251-292.

**Torus orbits in G/P.** / Flaschka, Hermann; Haine, Luc.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 149, no. 2, pp. 251-292.

}

TY - JOUR

T1 - Torus orbits in G/P

AU - Flaschka, Hermann

AU - Haine, Luc

PY - 1991

Y1 - 1991

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=84974000313&partnerID=8YFLogxK

M3 - Article

VL - 149

SP - 251

EP - 292

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -