Recent Developments in Noncommutative Algebraic Geometry: "Quasi-flag manifolds and moment graphs"

April 9, 2024
  • 14A22
We will discuss a new class of topological G-spaces generalizing the classical flag manifolds of compact connected Lie groups. These spaces --- which we call quasi-flag manifolds --- are topological realizations of (derived) schemes of quasi-invariants of finite reflection groups. Many fundamental properties and geometric structures related to the flag manifolds can be extended to quasi-flag manifolds. In this talk, we will focus on categorical and homotopy-theoretic aspects: in particular, we will describe a new universal `gluing' construction of quasi-flag manifolds in terms of simplicially enriched moment graphs. This last construction is inspired by recent developments in abstract homotopy theory (Lurie's construction of the rigidification functor for simplicial sets and his straightening/unstraightening equivalence in the theory of infinity-categories). It applies to a broad class of spaces (known as GKM spaces) whose cohomological properties can be described combinatorially in terms of moment graphs. Time permitting, we will also discuss some applications in the context of representation theory and stable homotopy theory: in particular, we will look at topological analogues (spectral refinements) of basic algebraic properties of quasi-invariants such as the Gorenstein property. (Based on joint work with A. Ramadoss and Yun Liu).