Topological realization of algebras of quasi-invariants, I (with an Appendix by M. V. Feigin and K. E. Feldman)
Yuri Berest, Ajay C. Ramadoss
TL;DR
The paper develops a topological realization framework for algebras of quasi-invariants $Q_m(W)$ of Weyl groups, focusing on the rank-one case $G=SU(2)$ and constructing spaces $F_m(G,T)$ and their Borel quotients $X_m(G,T)$ that realize $Q_m(W)$ as cohomology rings. It establishes a Ganea fibre-cofibre tower to generate a sequence of quasi-flag manifolds and proves key identifications: $H^*(X_m( ext{SU}(2),T),Q)\cong Q_m(W)$, $K_G(F_m)\cong oldsymbol{Q}_m(W)$, and elliptic cohomology descriptions, along with a detailed analysis of divided-difference operators in this topological context. The work also explores 'fake' spaces arising from Rector spaces, showing that while they share rational cohomology, they differ in $K$-theory and $p$-local structure, thereby clarifying which constructions truly realize the quasi-invariant algebras. A topological Gorenstein duality is established for the cochain spectra associated with $X_m$, providing a homotopy-theoretic analogue of the classical Gorenstein property and extending the framework to elliptic cohomology and twisted theories. Overall, the paper lays foundational links between topology, representation theory, and generalized cohomology theories in the study of quasi-invariants.
Abstract
This is the first in a series of papers, where we introduce and study topological spaces that realize the algebras of quasi-invariants of finite reflection groups. Our result can be viewed as a generalization of a well-known theorem of A. Borel that realizes the ring of invariant polynomials a Weyl group $W$ as a cohomology ring of the classifying space $BG$ of the associated Lie group $G$. In the present paper, we state our realization problem for the algebras of quasi-invariants of Weyl groups and give its solution in the rank one case (for $G = SU(2)$). We call the resulting $G$-spaces $ F_m(G,T) $ the $m$-quasi-flag manifolds and their Borel homotopy quotients $ X_m(G,T) $ the spaces of $m$-quasi-invariants. We compute the equivariant $K$-theory and the equivariant (complex analytic) elliptic cohomology of these spaces and identify them with exponential and elliptic quasi-invariants of $W$. We also extend our construction of spaces quasi-invariants to a certain class of finite loop spaces $ ΩB $ of homotopy type of $ S^3 $ originally introduced by D. L. Rector. We study the cochain spectra $ C^*(X_m,k) $ associated to the spaces of quasi-invariants and show that these are Gorenstein commutative ring spectra in the sense of Dwyer, Greenlees and Iyengar.