Cellular homology of real flag manifolds

Let ${\mathbb{F}}_{\Theta }=G∕P_{\Theta }$ be a generalized flag manifold, where $G$ is a real non-compact semi-simple Lie group and $P_{\Theta }$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow ${\mathbb{F}}_{\Theta }$ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in ${\mathbb{R}}^n$ and use them to compute the boundary operator $\partial$ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of $\partial$ are 0 or $\pm 2$ (so that ${\mathbb{Z}}_2$-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.