Torus orbits in G/P

Hermann Flaschka, Luc Haine

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23 Scopus citations

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

ASJC Scopus subject areas

  • Mathematics(all)

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