Quasi-flag manifolds and moment graphs

TL;DR

This work introduces and studies m-quasi-flag manifolds F_m(G,T), showing their even rational G–equivariant cohomology is naturally isomorphic to the k-quasi-invariants Q_k(W) with k determined by m, and that they assemble into a W–equivariant diagram over the multiplicity set ℳ(W). It develops two parallel realizations: (i) algebraic-geometric models via schemes and coaffine stacks (with V_k(W) and ˜V_m(W) and their coaffine counterparts) and (ii) topological decompositions over moment graphs (via the Bruhat graph Γ and gW-diagrams), using m-simplicial thickenings and plus constructions to produce simply connected models F_m^+(G,T). The paper proves freeness, Cohen–Macaulay properties, and natural module structures (nil-Hecke for odd m and Cherednik for even m) on ˜Q_m(W) and links these to K-theory through exponential quasi-invariants; it also extends the framework to GKM spaces with multiplicities. Collectively, these results integrate topology, invariant theory, and derived algebraic geometry to provide a unified, computationally accessible theory for quasi-invariants and their topological realizations, with broad potential for generalization to other GKM spaces.

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.

Figures (7)

• Figure 1: A 'thickened' planar graph
• Figure 2: The Bruhat graph $\Gamma$ for $W = S_3$
• Figure 3: The moment categories $\mathscr{C}^{(0)}(\Gamma)$, $\, \mathscr{C}^{(1)}(\Gamma)$ and $\mathscr{C}^{(2)}(\Gamma)$ for $W = \mathbb{Z}/2\mathbb{Z}$.
• Figure 4: The skeleton of $\mathscr{C}(\Gamma)_{hW}$
• Figure 5: The space $S\bigl(\mathbb O_{e(s_{\alpha}, w)}^{\ast (m_{\alpha}+1)}\bigr)$

Theorems & Definitions (175)

• Proposition 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Theorem 2.4
• proof
• Remark 2.5
• Proposition 2.6
• proof
• Theorem 2.7