I've frequently faced this deduction regarding reduction of structure groups to their maximal compact subgroups. Since each Lie group is product of its maximal compact subgroup and some contractible space it is homotopic to its maximal compact subgroup but I can't see the equivalence between this reduction and the choice of definite sub-bundles. and here I give this in context of generalized complex geometry.
We know that a a generalized complex structure $\mathcal{J}$ on a smooth manifold $M$ is a reduction of structure group of $TM\oplus T^*M$ to $U(n,n)$, we may be further reduced to $U(n)\times U(n)$. This corresponds geometrically to the choice of a positive definite sub-bundle $C_+ \subset TM⊕T^*M$ which is complex with respect to $\mathcal{J}$ . The orthogonal complement $C_−=C_+^{\perp}$ is negative-definite and also complex, and so we obtain the orthogonal decomposition $TM\oplus T^*M=C_+\oplus C_-$
precisely I have 2 questions: 1) what is the equivalence between reduction and this choice? 2)I know that generalized complex structure $\mathcal{J}:TM\oplus T^*M\to TM\oplus T^*M$ has the property that $\mathcal{J}^2=-1$ but why this induces complex structures on $C_{\pm}$?