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


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