Topological realization of algebras of quasi-invariants, I (with an Appendix by M. V. Feigin and K. E. Feldman)

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 $C^{\ast }(X_m,k)$ associated to the spaces of quasi-invariants and show that these are Gorenstein commutative ring spectra in the sense of Dwyer, Greenlees and Iyengar.

The paper has been substantially revised. The current version includes an Appendix (written by M. Feigin), which is based on an unpublished preprint of M. Feigin and K. Feldman