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- W4377130673 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 $ Omega B $ 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." @default.
- W4377130673 created "2023-05-21" @default.
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- W4377130673 date "2023-05-17" @default.
- W4377130673 modified "2023-09-29" @default.
- W4377130673 title "Topological realization of algebras of quasi-invariants, I" @default.
- W4377130673 doi "https://doi.org/10.48550/arxiv.2305.10604" @default.
- W4377130673 hasPublicationYear "2023" @default.
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