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On spaces of commuting elements in Lie groups†

Published online by Cambridge University Press:  12 May 2016

FREDERICK R. COHEN  and
MENTOR STAFA
FREDERICK R. COHEN
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A. e-mail: fred.cohen@rochester.edu
MENTOR STAFA
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, U.S.A. e-mail: mentor.stafa@math.ethz.ch
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Abstract

The main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(ℤn , G). We show that this stabilised space of homomorphisms decomposes after suspending once with “summands” which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(ℤn , G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilised space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilised space also allows the description of the additive reduced homology of the individual spaces Hom(ℤn , G), with the order of the Weyl group inverted.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 381 - 407
Copyright
Copyright © Cambridge Philosophical Society 2016 

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Footnotes

†

With an appendix by Vic Reiner.

References

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