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Explicit Cayley triples in real forms of \(E_8\). (English) Zbl 1013.22003
There are two non-compact real forms of the complex Lie algebra of type \(E_8\), namely, \(E_{8(8)}\) and \(E_{8(-24)}\). Let \({\mathfrak g}_{\mathbb{R}}\) denote one of these forms. Let \(\sigma\) denote its Cartan involution and let \(G_{\mathbb{R}}\) denote its adjoint group. An \({\mathfrak s}{\mathfrak l}_2\) triple \(\{H,X,Y\}\) in \({\mathfrak g}_{\mathbb{R}}\) is called a Cayley triple if \(\sigma(H)= -H\), \(\sigma(X)= -Y\) and \(\sigma(Y)=-X\). There is a one-to-one correspondence between \(G_{\mathbb{R}}\)-nilpotent orbits on \({\mathfrak g}_{\mathbb{R}}\) and \(K_{\mathbb{R}}\)-orbits on the set of Cayley triples [see Proposition 1 of the author’s second paper (mentioned below)]. The correspondence is given in this way: For a Cayley triple \(\{H,X,Y\}\), the corresponding \(G_{\mathbb{R}}\)-orbit is the one generated by \(X\).
In this paper, for every \(G_{\mathbb{R}}\)-orbit, the author finds representatives of \(H\), \(X\) and \(Y\) for the corresponding Cayley triple. In order to achieve this, the author gives a basis of the Lie algebra where the Lie bracket and the action of \(\sigma\) on it are written down explicitly. Representatives of the Cayley triples are expressed as linear combinations of this basis.
This paper is a continuation of two previous papers of the author [ibid. 184, 231-255 (1998; Zbl 1040.17004); 191, 1-23 (1999; Zbl 1040.17006)], where the author determines Cayley triples of other real exceptional Lie algebras. The author also assumes most of the notations from these two papers. As mentioned by the author, the result in this paper will be useful in determining the closure ordering for nilpotent orbits of \({\mathfrak g}\). It will also benefit mathematicians working in representation theory of the exceptional groups and the orbit method.

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E60 Lie algebras of Lie groups
17B25 Exceptional (super)algebras
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