Bad Representations and Homotopy of Character Varieties
Annales Henri Lebesgue, Volume 5 (2022), pp. 93-140.

Metadata

Keywordscharacter variety, Borel-de Siebenthal subgroups, free group, homotopy groups, singularities

Abstract

Let G be a connected reductive complex affine algebraic group, and let 𝔛 r denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in 𝔛 r , extending a theorem of Richardson and proving a Mumford-type result about topological singularities; this resolves conjectures of Florentino–Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel–de Siebenthal subgroups. We then use the codimension bound to calculate higher homotopy groups of the smooth locus of 𝔛 r , proving conjectures of Florentino–Lawton–Ramras. Lastly, using the earlier analysis of Borel–de Siebenthal subgroups, we prove a conjecture of Sikora about centralizers of irreducible representations in Lie groups.


References

[AB83] Atiyah, Michael F.; Bott, Raoul The Yang–Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond., Ser. A, Volume 308 (1983) no. 1505, pp. 523-615 | MR: MR702806 | Zbl: 0509.14014

[AB94] A’Campo, Norbert; Burger, Marc Réseaux arithmétiques et commensurateur d’après G. A. Margulis, Invent. Math., Volume 116 (1994) no. 1-3, pp. 1-25 | Article | MR: 1253187 | Zbl: 0833.22014

[BdS48] Borel, Armand; de Siebenthal, Jean Sur les sous-groupes fermés connexes de rang maximum des groupes de Lie clos, C. R. Math. Acad. Sci. Paris, Volume 226 (1948), pp. 1662-1664 | MR: 0025475 | Zbl: 0030.34101

[BGPG08] Bradlow, Steven B.; García-Prada, Oscar; Gothen, Peter B. Homotopy groups of moduli spaces of representations, Topology, Volume 47 (2008) no. 4, pp. 203-224 | Article | MR: MR2416769 | Zbl: 1165.14028

[BL15] Biswas, Indranil; Lawton, Sean Fundamental group of moduli spaces of representations, Geom. Dedicata, Volume 178 (2015), pp. 135-141 | Article | MR: 3397486 | Zbl: 1331.14016

[BLR15] Biswas, Indranil; Lawton, Sean; Ramras, Daniel A. Fundamental groups of character varieties: surfaces and tori, Math. Z., Volume 281 (2015) no. 1-2, pp. 415-425 | Article | MR: 3384878 | Zbl: 1349.14042

[Bor91] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer, 1991 | Article | MR: 1102012 | Zbl: 0726.20030

[Bot56] Bott, Raoul An application of the Morse theory to the topology of Lie-groups, Bull. Soc. Math. Fr., Volume 84 (1956), pp. 251-281 | Article | Numdam | MR: 0087035 | Zbl: 0073.40001

[Bot59] Bott, Raoul The stable homotopy of the classical groups, Ann. Math., Volume 70 (1959), pp. 313-337 | Article | MR: 0110104 | Zbl: 0129.15601

[Bou05] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics, Springer, 2005 (Translated from the 1975 and 1982 French originals by Andrew Pressley) | MR: 2109105 | Zbl: 1139.17002

[BS58] Bott, Raoul; Samelson, Hans Applications of the theory of Morse to symmetric spaces, Am. J. Math., Volume 80 (1958), pp. 964-1029 | Article | MR: 0105694

[Che55] Chevalley, Claude Invariants of finite groups generated by reflections, Am. J. Math., Volume 77 (1955), pp. 778-782 | Article | MR: 0072877 | Zbl: 0065.26103

[Cos00] Coste, Michel An introduction to semialgebraic geometry, 2000 (Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.178.1934&rep=rep1&type=pdf)

[Dol82] Dolgachev, Igor Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) (Lecture Notes in Mathematics), Volume 956, Springer, 1982, pp. 34-71 | Article | MR: 704986 | Zbl: 0516.14014

[Dol03] Dolgachev, Igor Lectures on invariant theory, London Mathematical Society Lecture Note Series, 296, Cambridge University Press, 2003 | Article | MR: MR2004511 | Zbl: 1023.13006

[Dré04] Drézet, Jean-Marc Luna’s slice theorem and applications, Algebraic group actions and quotients. Notes of the 23rd autumn school in algebraic geometry, Wykno, Poland, September 3–10, 2000, Hindawi Publishing Corporation, Cairo, 2004, pp. 39-89 | MR: MR2210794 | Zbl: 1109.14307

[Dyn52] Dynkin, Evgeniĭ B. Maximal subgroups of the classical groups, Tr. Mosk. Mat. O.-va, Volume 1 (1952), pp. 39-166 | MR: 0049903 | Zbl: 0048.01601

[Eng75] Engelking, Ryszard Topologia ogólna, 47, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Biblioteka Matematyczna [Mathematics Library]) | MR: 0500779 | Zbl: 0373.54001

[FH91] Fulton, William; Harris, Joe Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer, 1991 | MR: MR1153249 | Zbl: 0744.22001

[FL12] Florentino, Carlos; Lawton, Sean Singularities of free group character varieties, Pac. J. Math., Volume 260 (2012) no. 1, pp. 149-179 | Article | MR: 3001789 | Zbl: 1264.14064

[FL14] Florentino, Carlos; Lawton, Sean Topology of character varieties of Abelian groups, Topology Appl., Volume 173 (2014), pp. 32-58 | Article | MR: 3227204 | Zbl: 1300.14045

[FL17] Florentino, Carlos; Lawton, Daniel A. Seanand Ramras Homotopy groups of free group character varieties, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 1, pp. 143-185 | MR: 3676044 | Zbl: 1403.14011

[Gel08] Gelander, Tsachik On deformations of F n in compact Lie groups, Isr. J. Math., Volume 167 (2008), pp. 15-26 | Article | MR: 2448015 | Zbl: 1171.22007

[Gou89] Goursat, Édouard Sur les substitutions orthogonales et les divisions régulières de l’espace, Ann. Sci. Éc. Norm. Supér., Volume 6 (1889), pp. 9-102 | Article | Zbl: 21.0530.01

[Gué18a] Guérin, Clément Bad irreducible subgroups and singular locus for character varieties in PSL(p,), Geom. Dedicata, Volume 195 (2018), pp. 23-55 | Article | MR: 3820494 | Zbl: 1418.20005

[Gué18b] Guérin, Clément Centralizers of irreducible subgroups in the projective linear group, J. Group Theory, Volume 21 (2018) no. 5, pp. 789-816 | Article | MR: 3849671 | Zbl: 1437.20027

[Hal15] Hall, Brian Lie groups, Lie algebras, and representations, Graduate Texts in Mathematics, 222, Springer, 2015 (An elementary introduction) | Article | MR: 3331229 | Zbl: 1316.22001

[Har80] Hardt, Robert M. Semi-algebraic local-triviality in semi-algebraic mappings, Am. J. Math., Volume 102 (1980), pp. 291-302 | Article | MR: 564475 | Zbl: 0465.14012

[Hil93] Hilbert, David Ueber die vollen Invariantensysteme, Math. Ann., Volume 42 (1893) no. 3, pp. 313-373 | Article | MR: 1510781 | Zbl: 25.0173.01

[Hof09] Hofmann, Kyle Roger Triangulation of locally semi-algebraic spaces (2009) (https://deepblue.lib.umich.edu/bitstream/handle/2027.42/63851/krhofman_1.pdf?sequence=1&isAllowed=y) (Ph. D. Thesis) | MR: 2713888

[HP04] Heusener, Michael; Porti, Joan The variety of characters in PSL 2 (), Bol. Soc. Mat. Mex., Volume 10 (2004) no. Special Issue, pp. 221-237 | MR: MR2199350 | Zbl: 1100.57014

[Hum95] Humphreys, James E. Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, 43, American Mathematical Society, 1995 | MR: 1343976 | Zbl: 0834.20048

[JM87] Johnson, Dennis; Millson, John J. Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) (Progress in Mathematics), Volume 67, Birkhäuser, 1987, pp. 48-106 | Article | MR: 900823 | Zbl: 0664.53023

[Kac68] Kachi, Hideyuki Homotopy groups of compact Lie groups E 6 ,E 7 and E 8 , Nagoya Math. J., Volume 32 (1968), pp. 109-139 | Article | MR: 0233924 | Zbl: 0159.24802

[KM99] Kachi, Hideyuki; Mimura, Mamoru Homotopy groups of compact exceptional Lie groups, Proc. Japan Acad., Volume 75 (1999) no. 4, pp. 47-49 | MR: 1701523 | Zbl: 0946.55007

[Kwu64] Kwun, Kyung Whan Uniqueness of the open cone neighborhood, Proc. Am. Math. Soc., Volume 15 (1964), pp. 476-479 | Article | MR: 161319 | Zbl: 0129.38204

[Lam58] Lambek, Joachim Goursat’s theorem and the Zassenhaus lemma, Can. J. Math., Volume 10 (1958), pp. 45-56 | Article | MR: 98138 | Zbl: 0079.25202

[Law19] Lawton, Sean Topological Singularities in Affine Varieties, MathOverflow, 2019 (https://mathoverflow.net/q/325764)

[LR15] Lawton, Sean; Ramras, Daniel A. Covering spaces of character varieties, New York J. Math., Volume 21 (2015), pp. 383-416 (With an appendix by Nan-Kuo Ho and Chiu-Chu Melissa Liu) | MR: 3358550 | Zbl: 1339.57002

[LS17] Lawton, Sean; Sikora, Adam S. Varieties of characters, Algebr. Represent. Theory, Volume 20 (2017) no. 5, pp. 1133-1141 | Article | MR: 3707908 | Zbl: 1400.14123

[LS19] Lawton, Sean; Sikora, Adam S. Varieties of Characters, 2019 (https://arxiv.org/abs/1604.02164)

[Lun73] Luna, Dominique Slices étalés, Sur les groupes algébriques (Mémoires de la Société Mathématique de France), Société Mathématique de France, 1973, pp. 81-105 | Article | Numdam | MR: MR0342523 | Zbl: 0286.14014

[Lun76] Luna, Dominique Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier, Volume 26 (1976) no. 1, pp. 33-49 | Article | MR: MR0423398 | Zbl: 0315.20039

[MFK94] Mumford, David B.; Fogarty, John C.; Kirwan, Frances C. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge., 34, Springer, 1994 | Article | MR: MR1304906 | Zbl: 0797.14004

[Mil68] Milnor, John Singular points of complex hypersurfaces, Annals of Mathematics Studies, 61, Princeton University Press; University of Tokyo Press, 1968 | MR: 0239612 | Zbl: 0184.48405

[Mil17] Milne, James S. Algebraic groups. The theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics, 170, Cambridge University Press, 2017 | Article | MR: 3729270 | Zbl: 1390.14004

[Mim67] Mimura, Mamoru The homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ., Volume 6 (1967), pp. 131-176 | Article | MR: 0206958 | Zbl: 0171.44101

[Miy67] Miyata, Takehiko Note on direct summands of modules, J. Math. Kyoto Univ., Volume 7 (1967), pp. 65-69 | Article | MR: 0214585 | Zbl: 0189.03702

[MT64a] Mimura, Mamoru; Toda, Hirosi Homotopy groups of SU (3), SU (4) and Sp (2), J. Math. Kyoto Univ., Volume 3 (1964), pp. 217-250 | Article | MR: 0169242 | Zbl: 0129.15404

[MT64b] Mimura, Mamoru; Toda, Hirosi Homotopy groups of symplectic groups, J. Math. Kyoto Univ., Volume 3 (1964), pp. 251-273 | Article | MR: 0169243 | Zbl: 0129.15405

[Mum61] Mumford, David B. The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math., Inst. Hautes Étud. Sci. (1961) no. 9, pp. 5-22 | Article | Numdam | MR: 153682 | Zbl: 0108.16801

[Mum99] Mumford, David The red book of varieties and schemes, Lecture Notes in Mathematics, 1358, Springer, 1999 includes the Michigan lectures (1974) on curves and their Jacobians. With contributions by Enrico Arbarello | Article | MR: 1748380 | Zbl: 0945.14001

[Nag64] Nagata, Masayoshi Invariants of a group in an affine ring, J. Math. Kyoto Univ., Volume 3 (1964), pp. 369-377 | MR: MR0179268 | Zbl: 0146.04501

[NR69] Narasimhan, Mudumbai S.; Ramanan, Sundararaman Moduli of vector bundles on a compact Riemann surface, Ann. Math., Volume 89 (1969), pp. 14-51 | Article | MR: MR0242185 | Zbl: 0186.54902

[OV90] Onishchik, Arkadiĭ L.; Vinberg, Èrnest B. Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer, 1990 (translated from the Russian and with a preface by D. A. Leites) | Article | MR: 1064110 | Zbl: 0722.22004

[Pea75] Pears, Alan R. Dimension theory of general spaces, Cambridge University Press, 1975 | MR: 0394604 | Zbl: 0312.54001

[Rac74] Racine, Michel L. On maximal subalgebras, J. Algebra, Volume 30 (1974), pp. 155-180 | Article | MR: 0349771 | Zbl: 0282.17009

[Ram08] Ramras, Daniel A. Yang–Mills theory over surfaces and the Atiyah-Segal theorem, Algebr. Geom. Topol., Volume 8 (2008) no. 4, pp. 2209-2251 | Article | MR: 2465739 | Zbl: 1240.19005

[Ric88] Richardson, Roger W. Conjugacy classes of n-tuples in Lie algebras and algebraic groups, Duke Math. J., Volume 57 (1988) no. 1, pp. 1-35 | Article | MR: 952224 | Zbl: 0685.20035

[Rub92] Rubenthaler, Hubert Une classification des paires duales dans les algèbres de Lie réductives, C. R. Math. Acad. Sci. Paris, Volume 315 (1992) no. 6, pp. 645-648 | MR: 1183795 | Zbl: 0774.17011

[Sch04] Schwarz, Gerald W. Group actions and quotients for compact Lie groups and algebraic groups, Invariant theory in all characteristics. Proceedings of the workshop on invariant theory, Queen’s University, Kingston, ON, Canada, April 8–19, 2002 (CRM Proceedings & Lecture Notes), Volume 35, American Mathematical Society, 2004, pp. 209-227 | MR: 2066469 | Zbl: 1131.14132

[Sik12] Sikora, Adam S. Character varieties, Trans. Am. Math. Soc., Volume 364 (2012) no. 10, pp. 5173-5208 | Article | MR: 2931326 | Zbl: 1291.14022

[Sik15] Sikora, Adam S. G-character varieties for G=SO(n,) and other not simply connected groups, J. Algebra, Volume 429 (2015), pp. 324-341 | Article | MR: 3320627 | Zbl: 1330.14083

[ST54] Shephard, G. C.; Todd, J. A. Finite unitary reflection groups, Can. J. Math., Volume 6 (1954), pp. 274-304 | Article | MR: 0059914 | Zbl: 0055.14305

[Tit55] Tits, Jacques Sous-algèbres des algèbres de Lie semi-simples, Séminaire Bourbaki : années 1954/55 - 1955/56, exposés 101-136 (Séminaire Bourbaki), Volume 3, Société Mathématique de France, 1955, 119, pp. 197-214 (d’apr’ès V. Morozov, A. Malčev, E. Dynkin et F. Karpelevitch) | MR: 1611385

[vdBS59] van der Blij, Frederik; Springer, Tonny A. The arithmetics of octaves and of the group G 2 , Nederl. Akad. Wet., Proc., Volume 62 (1959), pp. 406-418 | MR: 0152555 | Zbl: 0089.25803