Strong involutions in finite special linear groups of odd characteristic

Let t be an involution in GL(n,q) whose fixed point space E+ has dimension k between n/3 and 2n/3. For each g GL(n,q) such that ttg has even order, 〈ttg〉 contains a unique involution z(g) which commutes with t. We prove that, with probability at least c/log n (for some c>0), the restriction z(g)|E+ is an involution on E+ with fixed point space of dimension between k/3 and 2k/3. This result has implications in the analysis of the complexity of recognition algorithms for finite classical groups in odd characteristic. We discuss how similar results for involutions in other finite classical groups would solve a major open problem in our understanding of the complexity of constructing involution centralisers in those groups.