Parabolic Homotopy Colimits and Coxeter Descents

Abstract

Let $G$ be a compact, connected, simply connected semisimple Lie group with Weyl group $W$ and simple reflections $S$. For a simplicial complex $\mathcal K$ on $S$, form the homotopy colimit $X_{\mathcal K}(G)=\operatorname*{hocolim}_{I\in\mathcal K}G/G_I$ of standard partial flag manifolds. We compute its integral homology. If $\operatorname{Des}_R(w)$ is the right descent set of $w\in W$ and $\ell(w)$ its Coxeter length, then $$ H_n(X_{\mathcal K}(G);\mathbb Z)\cong \bigoplus_{w\in W}\widetilde H_{n-2\ell(w)-1}(\mathcal K_{\operatorname{Des}_R(w)};\mathbb Z). $$ Thus the induced subcomplexes of $\mathcal K$ supply the topological data, while the Weyl group determines which subcomplex occurs and the Schubert-degree shift. The proof gives a chain-level splitting and an integral Morse reduction. We derive homotopy detection, duality and rigidity results, and recover polyhedral products, matroid--Tutte formulas, and the adjoint sphere as special cases.