Semigroup actions on homogeneous spaces |
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Authors: | Luiz A B San Martin Pedro A Tonelli |
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Institution: | 1. Instituto de Matemática, Estatística e Ciência da Computa??o, Universidade Estadual de Campinas, Cx. Postal 6065, 13084-100, Campinas, S.P., Brasil 2. Instituto de Matemática e Estatística, Universidade de S?o Paulo, Cx. Postal 20570, 01452-990, S?o Paulo, S. P.
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Abstract: | LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD
w
.D
1 is the only invariant control set, and the subsetW(S)={w:D
w
=D
1} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold.
Research partially supported by CNPq grant no 50.13.73/91-8 |
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