Reality properties of conjugacy classes in algebraic groups

ISSN： 0021-2172

Source： Israel Journal of Mathematics, Vol.165, Iss.1, 2008-06, pp. : 1-27

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Abstract

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, h ² = 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 G ₂ 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 G ₂.