**Journal of Lie Theory
**

Vol. 8, No. 2, pp. 335-350 (1998)

#
Order and domains of attraction of control sets in flag manifolds

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L. San Martin

Instituto de Matematica

Universidade Estadual de Campinas

Cx. Postal 6065

13.081-970 Campinas, SP, Brasil

smartin@ime.unicamp.br

**Abstract:** Let $G$ be a real semi-simple noncompact Lie group and $S\subset G$ a subsemigroup with $\inte S\neq \emptyset $. This article relates the Bruhat-Chevalley order in the Weyl group $W$ of $G$ to the ordering of the control sets for $S$ in the flag manifolds of $G$ by showing that the one-to-one correspondence between the control sets and the elements of a double coset $W\left( S\right) \backslash W/W_{\Theta }$ of $W$ reverses the orders. This fact is used to show that the domain of attraction of a control set is a union of Schubert cells.

**Keywords:** semigroups, semi-simple groups, flag manifolds, control sets, Bruhat-Chevalley order

**Classification (MSC91):** 20M20, 54H15; 93B

**Full text of the article:**

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