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Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 591–610 | Cite as

Hilbert–Poincaré series for spaces of commuting elements in Lie groups

  • Daniel A. RamrasEmail author
  • Mentor Stafa
Article

Abstract

In this article we study the homology of spaces \(\mathrm{Hom}({\mathbb Z}^n,G)\) of ordered pairwise commuting n-tuples in a Lie group G. We give an explicit formula for the Poincaré series of these spaces in terms of invariants of the Weyl group of G. By work of Bergeron and Silberman, our results also apply to \(\mathrm{Hom}(F_n/\Gamma _n^m,G)\), where the subgroups \(\Gamma _n^m\) are the terms in the descending central series of the free group \(F_n\). Finally, we show that there is a stable equivalence between the space \(\mathrm{Comm}(G)\) studied by Cohen–Stafa and its nilpotent analogues.

Keywords

Representation space Hilbert–Poincaré series Characteristic degree Finite reflection group 

Mathematics Subject Classification

Primary 22E99 55N10 Secondary 20F55 57T10 

Notes

Acknowledgements

We thank Alejandro Adem and Fred Cohen for helpful comments, and Mark Ramras for pointing out the Binomial Theorem, which simplified our formulas.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Indiana University, Purdue University IndianapolisIndianapolisUSA
  2. 2.Tulane UniversityNew OrleansUSA

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