2023/05/17 by Yuri Berest, Berest, Yuri, Ajay C. Ramadoss +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2305.10604
openalex publication_date 2023/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Fm(G,T) the m-quasi-flag manifolds and their Borel homotopy quotients Xm(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 S3 originally introduced by D. L. Rector. We study the cochain spectra C^*(Xm,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.