Reality of unipotent elements in classical Lie groups

Reversibility of affine transformations

“…Let g := (σ, y)(α, z)(σ, y) −1 . Then g is an involution in Aff(n, D) and g(A, v)g −1 = (A, v) −1 ; see [7,Lemma 2.1]. This completes the proof.…”

“…The adjoint action of a linear Lie group G on its Lie algebra g is given by the conjugation, i.e., Ad(g)X := gXg −1 . An element X ∈ g is called Ad G -real if −X = gXg −1 for some g ∈ G. An Ad G -real element X ∈ g is called strongly Ad Greal if −X = τ Xτ −1 for some involution τ ∈ G; see [7,Definition 1.1]. Observe that if −X = gXg −1 for some g ∈ G, then (exp(X)) −1 = g exp(X)g −1 .…”

“…In this section, we shall use the adjoint reality approach introduced in [7] to show that every element of with a unipotent linear part is strongly reversible. In view of Lemma 3.4 and Proposition 3.6, classification of reversible and strongly reversible elements in reduces to the case when the linear part of the affine group element is unipotent.…”

“…Now we recall the notion of adjoint reality for a linear Lie group G , which was introduced in [7]. The adjoint action of a linear Lie group G on its Lie algebra is given by the conjugation, i.e., .…”

“…We consider the Lie algebra of the affine group and consider the adjoint action; see Equation (3.4). Then we apply the notion of ‘adjoint reality’ introduced in [7], also see Section 3.3, to classify the strongly reversible elements in whose linear parts are unipotent; see Proposition 3.11.…”