Fixed point spaces in primitive actions of simple algebraic groups. (Q1400195)

Let \(G\) be a simple algebraic group over an algebraically closed field and \(M\) a maximal closed subgroup of \(G\) or \(M\) finite. Let \(\Omega=G/M\). One wants to study the size of the fixed points of an element of \(x\in G\) -- i.e. what is the codimension of the fixed point space \(\Omega(x)\) of \(x\) in \(\Omega\). This has applications to many kinds of problems. The main results in the article show that typically there is an involution in \(G\) with a reasonably large fixed point space. More precisely, unless \(M\) is finite of odd order or \(M\) is one of six explicit cases, then there is an involution \(x\) with \(\dim\Omega(x)/\dim\Omega\geq 1/2-1/(2h+1)\), where \(h\) is the Coexter number for \(G\). The proof is based on the classification of maximal subgroups of simple algebraic groups.