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Geodesic orbit metrics on compact simple Lie groups arising from flag manifolds. (Métriques définies par les variétés de drapeaux sur les groupes de Lie compacts, simples, dont les géodésiques sont des orbites.) (English. French summary) Zbl 1397.53065

Let \(\mathfrak{G}\) be a Lie group and \(\mathfrak{H}\) a closed subgrop of \(\mathfrak{G}\). A left \(\mathfrak{G}\)-invariant metric \(g\) on the homogeneous space \(\mathfrak{G} / \mathfrak{H}\) is a geodesic orbit metric if all the geodesics of \(g\) are depleted by the orbits of the \(1\)-parameter subgroups of \(\mathfrak{G}\). The left \(\mathfrak{G}\)-invariant metrics \(g\) on \(\mathfrak{G} / \mathfrak{H}\) are known to be in a bijective correspondence with the \(\text{Ad} (\mathfrak{H})\)-invariant inner products \((\text{ }, \text{ }) _g\) on the tangent space \(\mathfrak{M}\) to \(\mathfrak{G} / \mathfrak{H}\) at the origin. A left \(\mathfrak{G}\)-invariant metric \(g\) on \(\mathfrak{G} / \mathfrak{H}\) is naturally reductive if \(( [Z,X]_{\mathfrak{M}}, Y) _g + (X, [ Z, Y] _{\mathfrak{M}})_g =0\) for all \(X, Y, Z \in \mathfrak{M}\). The negative \(- B\) of the Killing form \(B\) of \(\text{Lie} ( \mathfrak{G})\) is associated with a naturally reductive metric on \(\mathfrak{G} / \mathfrak{H}\), which is called standard.
Let \(G\) be a compact simple Lie group, \(S \subset G\) be a torus and \(C(S)\) be the centralizer of \(S\) in \(G\). The homogeneous spaces of the form \(G / C(S)\) are called flag manifolds. The isometry group of \(G\) is of the form \(G \times K\) for some closed subgroup \(K\) of \(G\). A \(G \times K\)-invariant metric \(g\) on \(G = (G \times K) / K\) arises from a geodesic orbit metric on a flag manifold if \(K = C(S)\) is the centralizer of a torus \(S \subset G\) and the restriction \((\text{ }, \text{ }) _g | _{\mathfrak{M}}\) of the \(\text{Ad}(C(S))\)-invariant scalar product \((\text{ }, \text{ }) _g : \text{Lie} (G) \times \text{Lie} (G) \rightarrow {\mathbb R}\) on the Killing orthogonal complement \(\mathfrak{M}\) of \(\text{Lie} (C(S))\) to \(\text{Lie} (G)\) is associated with a left \(G\)-invariant geodesic orbit metric on \(G / C(S)\). The article under review shows that if a \(G \times C(S)\)-invariant metric \(g\) on a compact simple Lie group \(G\) arises from a geodesic orbit metric on the flag manifold \(G / C(S)\) then \(g\) is naturally reductive. The only \(G / C(S)\), which admit non-standard geodesic orbit metrics are \( \mathrm{SO} (2n+1) / \mathrm{U}(n)\) with \(n \geq 2\) and \( \mathrm{Sp} (n) / \mathrm{U}(1) \mathrm{Sp} (n-1)\) with \(n \geq 3\). The authors decompose the Lie algebra of the isotropy group \(C(S) = \mathrm{U}(n)\), respectively, \(C(S) = \mathrm{U}(1) \mathrm{Sp} (n-1)\) into a direct sum of its center \(u(1)\) and a simple Lie algebra \(\mathfrak{k}_o\). They use also that \(\mathfrak{M} = \text{Lie} (C(S)) ^{\perp} = \mathfrak{M}_1 \oplus \mathfrak{M}_2\) is a direct sum of two mutually inequivalent irreducible \(\text{Ad} C(S)\)-submodules \(\mathfrak{M}_1\), \(\mathfrak{M}_2\), in order to express the restrictions of \(( \text{ }, \text{ }) _g\) on \(u(1)\), \(\mathfrak{k}_o\), \(\mathfrak{M}_1\), \(\mathfrak{M}_2\) by the Killing form \(B\) of \(\text{Lie} (G)\) and to derive the natural reductiveness of \(g\).

MSC:

53C30 Differential geometry of homogeneous manifolds
14M17 Homogeneous spaces and generalizations
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:

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