Reality Properties of Conjugacy Classes in Algebraic Groups

Let G be an algebraic group defined over a field k. We call g ∈ G real if g is conjugate to g−1 and g ∈ G(k) as k-real if g is real in G(k). An element g ∈ G is strongly real if ∃h ∈ G, h2 = 1 (i.e. h is an involution) such that hgh−1 = g−1. Clearly, strongly real elements are real and are product of two involutions. Let G be a connected adjoint semisimple group over a perfect field k, with −1 in the Weyl group. We prove that any strongly regular k-real element in G(k) is strongly k-real (i.e. is a product of two involutions in G(k)). For classical groups, with some mild exceptions, over an arbitrary field k of characteristic not 2, we prove that k-real semisimple elements are strongly k-real. We compute an obstruction to reality and prove some results on reality specific to fields k with cd(k) ≤ 1. Finally, we prove that in a group G of type G2 over k, characteristic of k different from 2 and 3, any real element in G(k) is strongly k-real. This extends our results in [ST05], on reality for semisimple and unipotent real elements in groups of type G2.