Order and domains of attraction of control sets in flag manifolds

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

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