Mathematische Zeitschrift

, Volume 236, Issue 2, pp 251–290

Homology stability for classical regular semisimple varieties

  • G.I. Lehrer
  • G.B. Segal
Original article

DOI: 10.1007/PL00004831

Cite this article as:
Lehrer, G. & Segal, G. Math Z (2001) 236: 251. doi:10.1007/PL00004831

Abstract.

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group \(GL_n({\Bbb C})\). We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety.

Mathematics Subject Classification (1991): 57T10 22E10 55P35, 20G40 22E46 

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • G.I. Lehrer
    • 1
  • G.B. Segal
    • 2
  1. 1.School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia (e-mail: g.lehrer@maths.usyd.edu.au) AU
  2. 2.Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB2 1SB, England GB

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