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

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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.

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Notes

  1. Some authors (e.g. Grove and Benson [17, Chapter 7], or Humphreys [19, Chapter 3]) replace V by its dual \(V^*\) in this discussion.

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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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Authors and Affiliations

  1. Indiana University, Purdue University Indianapolis, Indianapolis, IN, 46202, USA

    Daniel A. Ramras

  2. Tulane University, New Orleans, LA, 70118, USA

    Mentor Stafa

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  1. Daniel A. Ramras
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  2. Mentor Stafa
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Correspondence to Daniel A. Ramras.

Additional information

D. Ramras was partially supported by a Simons Collaboration grant (#279007).

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Ramras, D.A., Stafa, M. Hilbert–Poincaré series for spaces of commuting elements in Lie groups. Math. Z. 292, 591–610 (2019). https://doi.org/10.1007/s00209-018-2122-1

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