Second cohomology of compact homogeneous spaces (Q2855497)

Let \(M\) be a compact simply connected semisimple real Lie group and let \(K\subset M\) be a closed subgroup. The connected component of \(K\) containing the identity element is denoted by \(K^0\). The center of the Lie algebra of \(K\) is denoted by \(\mathfrak{z}(\mathfrak{l})\). The adjoint action of \(K\) on the center of the Lie algebra of \(K\) factors through the quotient group \(K/K^0\). Let us consider the action of \(K/K^0\) on the dual space \(\mathfrak{z}(\mathfrak{l})^*\) corresponding to the adjoint action of \(K/K^0\) on \(\mathfrak{z}(\mathfrak{l})\). Let NEWLINE\[NEWLINE [\mathfrak{z}(\mathfrak{l})^*]^{K/K^0} \subset \mathfrak{z}(\mathfrak{l})^* NEWLINE\]NEWLINE be the space of invariants for this action.NEWLINENEWLINEThen the authors prove the following result using Hurewicz and Galois covering maps:NEWLINENEWLINEThe second real cohomology \(H^2 (M/K, \mathbb R)\) is canonically isomorphic to \([\mathfrak{z}(\mathfrak{l})^*]^{K/K^0}\).NEWLINENEWLINEThis isomorphism can be interpreted in the language of characters of closed group \(K\) as follows.NEWLINENEWLINELet us take any homomorphism \(\chi: K\rightarrow {\mathbb C}^*\). The group acts on \(M \times \mathbb C\) as follows: the action of any \(g\in K\) sends any \((m,\lambda)\in M\times \mathbb C\) to \((mg, \chi(g^{-1}) \lambda)\). The projection \((M\times \mathbb C)/ K \rightarrow M/K\) is a line bundle on \(M/K\) which is denoted by \(L^{\chi}\). Now \(\chi \rightarrow c_1(L^{\chi})\) produces a vector space homomorphism NEWLINE\[NEWLINE Char(K)\otimes_{\mathbb Z} \mathbb R \rightarrow H^2 (M/K,\mathbb R), NEWLINE\]NEWLINE where \(Char(K)\) is the group of characteristics of \(K\). Onthe other hand we have NEWLINE\[NEWLINE Char(K)\otimes_{\mathbb Z} \mathbb R = [\mathfrak{z}(\mathfrak{l})^*]^{K/K^0}. NEWLINE\]NEWLINE So we have this canonical isomorphism.