Quasi-flag manifolds and moment graphs

Abstract

We introduce and study a new class of topological $G$-spaces generalizing the classical flag manifolds $G/T$ of compact connected Lie groups. These spaces, which we call the $m$-quasi-flag manifolds $F_m=F_m(G,T)$, are topological realizations of the algebras $Q_k(W)$ of $k$-quasi-invariant polynomials of the Weyl group $W$ in the sense that their (even-dimensional) $G$-equivariant cohomology $H_G(F_m,\mathbb{C})$ is naturally isomorphic to $Q_k(W)$, where $m$ is a $W$-invariant integer-valued multiplicity function on the system of roots of $W$ and $k=\frac{m}{2}$ or $\frac{m+1}{2}$ depending on whether $m$ is even or odd. Many topological properties and algebraic structures related to the flag manifolds can be extended to quasi-flag manifolds. We compute the cohomology of quasi-flag manifolds by constructing their rational algebraic models in terms of coaffine stacks -- a certain kind of derived stacks introduced by B.Toën and J. Lurie to provide an algebro-geometric framework for rational homotopy theory. Besides cohomology, we also compute the equivariant K-theory of quasi-flag manifolds and extend some of our cohomological results to the multiplicative setting. On the topological side, our approach is strongly influenced by the classical work on homotopy decompositions of classifying spaces of compact Lie groups; however, the diagrams that we use in our decompositions do not arise from collections of subgroups of $G$ but rather from moment graphs -- combinatorial objects introduced in a different area of topology called the GKM theory.

A new substantially revised version. Added: 1) explicit examples and computations of rings of quasi-covariants (Example 1.1 and 1.2); a new geometric model for thickened moment graphs in terms of spherical polyhedral complexes (Section 5.2); a new appendix (Appendix C) on homotopy decompositions of infinity-categories; 136 pages; 8 figures