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


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.


Representation space Hilbert–Poincaré series Characteristic degree Finite reflection group 

Mathematics Subject Classification

Primary 22E99 55N10 Secondary 20F55 57T10 



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


  1. 1.
    Adem, A., Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: On decomposing suspensions of simplicial spaces. Bol. Soc. Mat. Mexicana (3) 15(1), 91–102 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Adem, A., Cohen, F.R.: Commuting elements and spaces of homomorphisms. Math. Ann. 338(3), 587–626 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adem, Alejandro, Gómez, José Manuel: Equivariant \(K\)-theory of compact Lie group actions with maximal rank isotropy. J. Topol. 5(2), 431–457 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baird, T.: Cohomology of the space of commuting \(n\)-tuples in a compact Lie group. Algebr. Geom. Topol. 7, 737–754 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baird, T., Jeffrey, L.C., Selick, P.: The space of commuting \(n\)-tuples in SU\((2)\). Illinois J. Math. 55(3), 805–813 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bergeron, M.: The topology of nilpotent representations in reductive groups and their maximal compact subgroups. Geom. Topol. 19, 1383–1407 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bergeron, M., Silberman, L.: A note on nilpotent representations. J. Group Theory 19(1), 125–135 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogenes de groupes de Lie compacts. Ann. Math. 57(1), 115–207 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borel, A, Friedman, R., Morgan, J.: Almost Commuting Elements in Compact Lie Groups. Number 747 in Mem. Amer. Math. Soc. AMS (2002)Google Scholar
  10. 10.
    Bott, R., Samelson, H.: On the Pontryagin product in spaces of paths. Comment. Math. Helv. 27(1), 320–337 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Broué, M.: Introduction to complex reflection groups and their braid groups, volume 1988 of Lecture Notes in Mathematics, Springer, Berlin (2010)Google Scholar
  12. 12.
    Brown, Ronald: Topology and groupoids. BookSurge LLC, Charleston (2006)zbMATHGoogle Scholar
  13. 13.
    Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math. 77(4), 778–782 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cohen, F. R., Stafa, M.: A survey on spaces of homomorphisms to Lie groups. In: Configurations Spaces: Geometry, Topology and Representation Theory, volume 14 of Springer INdAM series, Springer, pp. 361–379 (2016)Google Scholar
  15. 15.
    Cohen, F.R., Stafa, M.: On spaces of commuting elements in Lie groups. Math. Proc. Camb. Philos. Soc. 161(3), 381–407 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gómez, J., Pettet, A., Souto, J.: On the fundamental group of \({\rm Hom}({\mathbb{Z}}^k, G)\). Math. Z. 271(1–2), 33–44 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grove, L.C., Benson, C.T.: Finite reflection groups, volume 99 of Graduate Texts in Mathematics, second edition. Springer, New York (1985)Google Scholar
  18. 18.
    Halperin, S.: Rational homotopy and torus actions, pages 293–306. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1985)Google Scholar
  19. 19.
    Humphreys, J.E.: Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29. Reprint edition. Cambridge University Press, Cambridge (1992)Google Scholar
  20. 20.
    Kac, V., Smilga, A.: Vacuum structure in supersymmetric Yang-Mills theories with any gauge group. The many faces of the superworld, pp. 185–234 (2000)Google Scholar
  21. 21.
    Molien, Th: Ueber die Invarianten der linearen Substitutionsgruppen. Berl. Ber. 1152–1156, 1897 (1897)zbMATHGoogle Scholar
  22. 22.
    Pettet, A., Souto, J.: Commuting tuples in reductive groups and their maximal compact subgroups. Geom. Topol. 17(5), 2513–2593 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Reeder, M.: On the cohomology of compact Lie groups. Enseign. Math. 41, 181–200 (1995)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Richardson, R.W.: Commuting varieties of semisimple Lie algebras and algebraic groups. Compositio Math. 38(3), 311–327 (1979)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math 6(2), 274–301 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sjerve, D., Torres-Giese, E.: Fundamental groups of commuting elements in Lie groups. Bull. Lond. Math. Soc. 40(1), 65–76 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Springer, T.A.: Regular elements of finite reflection groups. Invent. Math. 25(2), 159–198 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stafa, M.: On polyhedral products and spaces of commuting elements in lie groups. PhD thesis, University of Rochester (2013)Google Scholar
  29. 29.
    Stafa, M.: Poincaré series of character varieties for nilpotent groups. arXiv preprint arXiv:1705.01443 (2017)
  30. 30.
    Villarreal, B.: Cosimplicial groups and spaces of homomorphisms. Algebra Geom. Topol. 17(6), 3519–3545 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Witten, E.: Constraints on supersymmetry breaking. Nuclear Phys. B 202, 253–316 (1982)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Witten, E.: Toroidal compactification without vector structure. J. High Energy Phys., (2):Paper 6, 43 (1998)Google Scholar

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

Personalised recommendations