On the signature of homogeneous spaces (Q1194379)

The author calculates the signatures of real and quaternionic Grassmannians and all homogeneous spaces of compact exceptional Lie groups. Let \(G\), \(H\) be compact connected Lie groups such that \(H\subset G\) and \(\text{rank}(G)=\text{rank}(H)\). Relative to a common maximal torus \(T\subset H\subset G\) one denotes by \(\Sigma\) resp. \(\Sigma'\) the root systems of \(T\) in \(G\) resp. \(H\), and by \(W\) resp. \(W'\) the corresponding Weyl groups. Let \(\Psi=\Sigma^ +\backslash(\Sigma')^ +\), where ``\(+\)'' denotes the corresponding sets of positive roots. The calculation of the signature is based on the general formula \[ \text{sign}(G/H)={1\over| W'|}\sum_{w\in W}(-1)^{\mu(w)}, \] where \(\mu(w)\) counts the number of complementary roots \(\gamma\in \Psi\) made negative by \(w^{-1}\). The numbers \(\mu(w)\) are calculated separately for each case of \(\Psi\) by combinatorial methods. The paper is an appendix to \textit{F. Hirzebruch} and \textit{P. Slodowy} [ibid. 35, No. 1-3, 309-343 (1990; Zbl 0712.57010)]. The latter contains basic definitions, motivations, the above formula for the signature, calculations of \(\text{sign}(G/H)\) for other symmetric spaces and the main equality \(\phi(X)=\text{sign}(X)\) for all connected oriented homogeneous spaces admitting a spin structure \((\phi(X)\) denotes the normalized elliptic genus of a homogeneous space \(X\)).