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Note on Hirzebruch’s proportionality principle. (English) Zbl 0726.57019
Let G be a connected semisimple real Lie group, H its closed subgroup and $${\mathfrak g}$$, $${\mathfrak h}$$ the corresponding Lie algebras. The pair (G,H) is called $$\theta$$-stable if there exists a Cartan involution $$\theta$$ of G such that $$\theta (H)=H$$ and if the connected Lie subgroup $$H_{{\mathbb{C}}}$$ of $$G_{{\mathbb{C}}}=Int({\mathfrak g}\otimes {\mathbb{C}})$$ corresponding to $${\mathfrak h}\otimes {\mathbb{C}}$$ is closed. One associates with a $$\theta$$-stable pair (G,H) the homogeneous spaces G/H, $$G_{{\mathbb{C}}}/H_{{\mathbb{C}}}$$ and $$G_ U/H_ U$$, where $$G_ U$$, $$H_ U$$ are compact real forms of $$G_{{\mathbb{C}}}$$, $$H_{{\mathbb{C}}}$$. Let $$\Gamma$$ be a discrete subgroup of G acting on G/H freely and properly discontinuously. The authors construct a $${\mathbb{C}}$$-algebra homomorphism $$H^*(G_ U/H_ U,{\mathbb{C}})\to H^*(\Gamma \setminus G/H,{\mathbb{C}})$$ which is injective if $$\Gamma\setminus G/H$$ is compact and H connected. Being given finite dimensional real linear representations of G and $$G_ U$$ having the same complexification, one constructs the associated vector bundles $$^{\Gamma}E\to \Gamma \setminus G/H$$ and $$E_ U\to G_ U/H_ U$$. It is proved that any polynomial relation among the real Pontryagin classes of $$E_ U$$ implies the same relation among those of $$^{\Gamma}E$$. The converse is true if $$\Gamma\setminus G/H$$ is compact and H connected. A similar property of the Chern classes is proved, too.
Reviewer: A.L.Onishchik

##### MSC:
 57R20 Characteristic classes and numbers in differential topology 53C30 Differential geometry of homogeneous manifolds