Quasi-flag manifolds, moment graphs and homotopy Lie groups
组织者
演讲者
Yuri Berest
时间
2025年04月03日 13:00 至 14:30
地点
A3-4-301
线上
Zoom 242 742 6089
(BIMSA)
摘要
In this talk, we will introduce 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 $F_m(G,T)$ --- are topological realizations of the rings $Q_m(W)$ of m-quasi-invariant polynomials of finite reflection groups, where $W$ is the Weyl group associated to $(G,T)$. Many properties and geometric structures related to the classical flag manifolds can be extended to quasi-flag manifolds (e.g. equivariant $K$-theory and elliptic cohomology can be computed for quasi-flag manifolds in an explicit way).
In general, the spaces $F_m(G,T)$ can be obtained from $G/T$ by a topological `gluing' construction that we call the m-simplicial thickening. This construction applies to other spaces than $G/T$: e.g., a partial flag manifold, $G/P$, or more generally, an arbitrary GKM manifold M with $G$-action. In this last case, the role of a Weyl group --- or rather, its root system --- is played by the moment graph $\Gamma$ of $M$, to which we can now associate a generalized ring $Q_m(\Gamma)$ of quasi-invariants. Another, perhaps more intriguing generalization, arises when we pass to the $p$-local setting ($p$ a fixed prime) and replace compact Lie groups with p-compact groups (aka homotopy Lie groups). In this case, we obtain topological realizations of algebras of quasi-invariants for some non-Coxeter (complex) reflection groups.
In general, the spaces $F_m(G,T)$ can be obtained from $G/T$ by a topological `gluing' construction that we call the m-simplicial thickening. This construction applies to other spaces than $G/T$: e.g., a partial flag manifold, $G/P$, or more generally, an arbitrary GKM manifold M with $G$-action. In this last case, the role of a Weyl group --- or rather, its root system --- is played by the moment graph $\Gamma$ of $M$, to which we can now associate a generalized ring $Q_m(\Gamma)$ of quasi-invariants. Another, perhaps more intriguing generalization, arises when we pass to the $p$-local setting ($p$ a fixed prime) and replace compact Lie groups with p-compact groups (aka homotopy Lie groups). In this case, we obtain topological realizations of algebras of quasi-invariants for some non-Coxeter (complex) reflection groups.