Topological realization of algebras of quasi-invariants, I (with an Appendix by M. V. Feigin and K. E. Feldman)
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908833855569920 |
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| author | Berest, Yuri Ramadoss, Ajay C. |
| author_facet | Berest, Yuri Ramadoss, Ajay C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_10604 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Topological realization of algebras of quasi-invariants, I (with an Appendix by M. V. Feigin and K. E. Feldman) Berest, Yuri Ramadoss, Ajay C. Algebraic Topology Representation Theory 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. |
| title | Topological realization of algebras of quasi-invariants, I (with an Appendix by M. V. Feigin and K. E. Feldman) |
| topic | Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2305.10604 |