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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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