Torus orbits in G/P

Hermann Flaschka, Luc Haine

Research output: Contribution to journalArticle

22 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)251-292
Number of pages42
JournalPacific Journal of Mathematics
Volume149
Issue number2
StatePublished - 1991

Fingerprint

Torus
Orbit
Divisor
Symplectic Geometry
Genericity
Parabolic Subgroup
Invariant Tori
Semisimple Lie Group
Algebraic Variety
Convex Body
Projective Space
Multiplicity
Closure
Correspondence
Momentum
Intersection
Linear Systems
Subgroup
Generator

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Flaschka, H., & Haine, L. (1991). Torus orbits in G/P. Pacific Journal of Mathematics, 149(2), 251-292.

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

In: Pacific Journal of Mathematics, Vol. 149, No. 2, 1991, p. 251-292.

Research output: Contribution to journalArticle

Flaschka, H & Haine, L 1991, 'Torus orbits in G/P', Pacific Journal of Mathematics, vol. 149, no. 2, pp. 251-292.
Flaschka, Hermann ; Haine, Luc. / Torus orbits in G/P. In: Pacific Journal of Mathematics. 1991 ; Vol. 149, No. 2. pp. 251-292.
@article{7d8472993c6948879e699e63a7904425,
title = "Torus orbits in G/P",
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.",
author = "Hermann Flaschka and Luc Haine",
year = "1991",
language = "English (US)",
volume = "149",
pages = "251--292",
journal = "Pacific Journal of Mathematics",
issn = "0030-8730",
publisher = "University of California, Berkeley",
number = "2",

}

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.

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

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

M3 - Article

AN - SCOPUS:84974000313

VL - 149

SP - 251

EP - 292

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -