Topology of character varieties of abelian groups. (English) Zbl 1300.14045
Let \(G\) be a complex reductive algebraic group. Let \(K\) be a maximal compact subgroup of \(G,\) and let \(\Gamma\) be a finitely generated abelian group of rank \(r\). Then \(G\) acts on \(\text{Hom}\left( \Gamma,G\right) ,\) and one can construct the affine GIT quotient \(\mathfrak{X}_{\Gamma}\left( G\right) =\text{Hom}\left( \Gamma,G\right) /\!/G,\) which is called the \(G\)-character variety of \(\Gamma.\) Additionally, \(G\) acts on \(\text{Hom}\left( \Gamma,K\right) ,\) giving rise to the GIT quotient \(\mathfrak{X}_{\Gamma}\left( K\right) ,\) similarly defined.
In the work under review, it is shown that there is a strong deformation retract from \(\mathfrak{X}_{\Gamma}\left( G\right) \) to \(\mathfrak{X} _{\Gamma}\left( K\right) .\) This result remains true upon restriction to the real locus \(\mathfrak{X}_{\Gamma}\left( G\left( \mathbb{R}\right) \right) .\) Additionally, it is shown that \(\mathfrak{X}_{\Gamma}\left( G\right) \) is irreducible if and only if \(\Gamma\) is a free abelian group, and either (1) \(r=1\); (2) \(r=2\) and \(G\) is simply connected; or (3) \(r\geq3\) and \(G\) is a product of special linear and compact symplectic groups.
In the work under review, it is shown that there is a strong deformation retract from \(\mathfrak{X}_{\Gamma}\left( G\right) \) to \(\mathfrak{X} _{\Gamma}\left( K\right) .\) This result remains true upon restriction to the real locus \(\mathfrak{X}_{\Gamma}\left( G\left( \mathbb{R}\right) \right) .\) Additionally, it is shown that \(\mathfrak{X}_{\Gamma}\left( G\right) \) is irreducible if and only if \(\Gamma\) is a free abelian group, and either (1) \(r=1\); (2) \(r=2\) and \(G\) is simply connected; or (3) \(r\geq3\) and \(G\) is a product of special linear and compact symplectic groups.
Reviewer: Alan Koch (Decatur)
MSC:
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14P25 | Topology of real algebraic varieties |
| 14L17 | Affine algebraic groups, hyperalgebra constructions |
| 14L24 | Geometric invariant theory |
| 22E46 | Semisimple Lie groups and their representations |
References:
| [1] | Adem, Alejandro; Cohen, Frederick R.; Gómez, José Manuel, Commuting elements in central products of special unitary groups, Proc. Edinb. Math. Soc., 56, 1, 1-12 (2013) · Zbl 1270.55013 |
| [2] | Adem, Alejandro; Gómez, José Manuel, On the structure of spaces of commuting elements in compact lie groups, (Configuration Spaces: Geometry, Topology and Combinatorics. Configuration Spaces: Geometry, Topology and Combinatorics, CRM Series, vol. 14 (2013), Publ. Scuola Normale Superiore, Birkhäuser) · Zbl 1277.43011 |
| [3] | Atiyah, M. F.; Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci., 308, 1505, 523-615 (1983) · Zbl 0509.14014 |
| [4] | Baird, Thomas; Jeffrey, Lisa; Selick, Paul, The topology of nilpotent representations in reductive groups and their maximal compact subgroups (2013) |
| [5] | Baird, Thomas John, Cohomology of the space of commuting \(n\)-tuples in a compact Lie group, Algebr. Geom. Topol., 7, 737-754 (2007) · Zbl 1163.57026 |
| [6] | Bergeron, Maxime, The space of commuting \(n\)-tuples in \(SU(2) (2009)\) |
| [7] | Birkes, D., Orbits of linear algebraic groups, Ann. Math., 93, 3, 459-475 (1971) · Zbl 0212.36402 |
| [8] | Borel, Armand, Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0726.20030 |
| [9] | Borel, Armand; Friedman, Robert; Morgan, John W., Almost commuting elements in compact Lie groups, Mem. Am. Math. Soc., 157, 747 (2002), x+136 · Zbl 0993.22002 |
| [10] | Bradlow, Steven B.; García-Prada, Oscar; Gothen, Peter B., Surface group representations and \(U(p, q)\)-Higgs bundles, J. Differ. Geom., 64, 1, 111-170 (2003) · Zbl 1070.53054 |
| [11] | Bredon, Glen E., Introduction to Compact Transformation Groups, Pure Appl. Math., vol. 46 (1972), Academic Press: Academic Press New York · Zbl 0246.57017 |
| [12] | Casimiro, A. C.; Florentino, C.; Lawton, S.; Oliveira, A. G., Topology of moduli spaces of free group representations in real reductive groups (2013) |
| [13] | Conrad, Brian, Reductive group schemes (2011), (sga3 summer school, 2011) Available at · Zbl 1349.14151 |
| [14] | Cooper, Daryl; Manning, Jason Fox, Non-faithful representations of surface groups into \(SL(2, C)\) which kill no simple closed curve · Zbl 1334.57003 |
| [15] | Dolgachev, Igor, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series, vol. 296 (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1023.13006 |
| [16] | Drazin, M. P., Some generalizations of matrix commutativity, Proc. Lond. Math. Soc., 3, 1, 222-231 (1951) · Zbl 0043.01701 |
| [17] | Florentino, Carlos; Casimiro, Ana Cristina, Stability of affine \(G\)-varieties and irreducibility in reductive groups, Int. J. Math., 23, 8, 30 (2012), 1250082 · Zbl 1258.14053 |
| [18] | Florentino, Carlos; Lawton, Sean, The topology of moduli spaces of free group representations, Math. Ann., 345, 2, 453-489 (2009) · Zbl 1200.14093 |
| [19] | Florentino, Carlos; Lawton, Sean, Character varieties and moduli of quiver representations, (The Tradition of Ahlfors-Bers, VI. The Tradition of Ahlfors-Bers, VI, Contemp. Math., vol. 590 (2013), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 9-38, 16Gxx (14L30) · Zbl 1325.16010 |
| [20] | García-Prada, O.; Riera, I. Mundet i., Representations of the fundamental group of a closed oriented surface in \(Sp(4, R)\), Topology, 43, 4, 831-855 (2004) · Zbl 1070.14014 |
| [21] | García-Prada, O.; Gothen, P. B.; Riera, I. Mundet i., The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations (2009) |
| [22] | Gel’fand, I. M.; Ponomarev, V. A., Remarks on the classification of a pair of commuting linear transformations in a finite-dimensional space, Funkc. Anal. Ego Prilož., 3, 4, 81-82 (1969) · Zbl 0204.45301 |
| [23] | Gerstenhaber, Murray, On dominance and varieties of commuting matrices, Ann. Math., 73, 324-348 (1961) · Zbl 0168.28201 |
| [24] | Gómez, José Manuel; Pettet, Alexandra; Souto, Juan, On the fundamental group of \(Hom(Z^k, G)\), Math. Z., 271, 1-2, 33-44 (2012) · Zbl 1301.55009 |
| [25] | Guralnick, Robert M., A note on commuting pairs of matrices, Linear Multilinear Algebra, 31, 1-4, 71-75 (1992) · Zbl 0754.15011 |
| [26] | Hatcher, Allen, Algebraic Topology (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1044.55001 |
| [27] | Hausel, Tamás; Thaddeus, Michael, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math., 153, 1, 197-229 (2003) · Zbl 1043.14011 |
| [28] | Hitchin, N. J., The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., 55, 1, 59-126 (1987) · Zbl 0634.53045 |
| [29] | Humphreys, James E., Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21 (1975), Springer-Verlag: Springer-Verlag New York · Zbl 0325.20039 |
| [30] | Kac, V. G.; Smilga, A. V., Vacuum structure in supersymmetric Yang-Mills theories with any gauge group, (The Many Faces of the Superworld (2000), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 185-234 · Zbl 1035.81061 |
| [31] | Kapustin, Anton; Witten, Edward, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys., 1, 1, 1-236 (2007) · Zbl 1128.22013 |
| [32] | Knapp, Anthony W., Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140 (2002), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 1075.22501 |
| [33] | Luna, D., Sur certaines opérations différentiables des groupes de Lie, Am. J. Math., 97, 172-181 (1975) · Zbl 0334.57022 |
| [34] | Luna, Domingo, Slices étales, (Sur les groupes algébriques. Sur les groupes algébriques, Bull. Soc. Math. France, Paris, Mémoire, vol. 33 (1973), Soc. Math.: Soc. Math. France, Paris), 81-105 · Zbl 0286.14014 |
| [35] | Luna, Domingo, Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier (Grenoble), 26, 1, 33-49 (1976), ix · Zbl 0315.20039 |
| [36] | Milne, James S., Basic theory of affine group schemes (2012), Available at |
| [37] | Mukai, Shigeru, An Introduction to Invariants and Moduli, Cambridge Studies in Advanced Mathematics, vol. 81 (2003), Cambridge University Press: Cambridge University Press Cambridge, translated from the 1998 and 2000 Japanese editions by W.M. Oxbury · Zbl 1033.14008 |
| [38] | Narasimhan, M. S.; Seshadri, C. S., Holomorphic vector bundles on a compact Riemann surface, (Differential Analysis. Differential Analysis, Bombay Colloq., 1964 (1964), Oxford Univ. Press: Oxford Univ. Press London), 249-250 · Zbl 0156.20401 |
| [39] | Pettet, Alexandra; Souto, Juan, Commuting tuples in reductive groups and their maximal compact subgroups, Geom. Topol., 17, 5, 2513-2593 (2013), 20G20 (55P99) · Zbl 1306.55007 |
| [40] | Richardson, R. W., Commuting varieties of semisimple Lie algebras and algebraic groups, Compos. Math., 38, 3, 311-327 (1979) · Zbl 0409.17006 |
| [41] | Richardson, R. W., Conjugacy classes of \(n\)-tuples in Lie algebras and algebraic groups, Duke Math. J., 57, 1, 1-35 (1988) · Zbl 0685.20035 |
| [42] | Richardson, R. W.; Slodowy, P. J., Minimum vectors for real reductive algebraic groups, J. Lond. Math. Soc., 42, 3, 409-429 (1990) · Zbl 0675.14020 |
| [43] | Schwarz, Gerald W., The topology of algebraic quotients, (Topological Methods in Algebraic Transformation Groups. Topological Methods in Algebraic Transformation Groups, New Brunswick, NJ, 1988. Topological Methods in Algebraic Transformation Groups. Topological Methods in Algebraic Transformation Groups, New Brunswick, NJ, 1988, Prog. Math., vol. 80 (1989), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 135-151 · Zbl 0708.14034 |
| [44] | Shafarevich, Igor R., Basic Algebraic Geometry, vol. 1 (1994), Springer-Verlag: Springer-Verlag Berlin, Varieties in projective space, translated from the 1988 Russian edition and with notes by Miles Reid · Zbl 0797.14001 |
| [45] | Shafarevich, Igor R., Basic Algebraic Geometry, vol. 2 (1994), Springer-Verlag: Springer-Verlag Berlin, Schemes and complex manifolds, translated from the 1988 Russian edition by Miles Reid · Zbl 0797.14002 |
| [46] | Sikora, Adam S., Character varieties of Abelian groups (2012) · Zbl 1291.14022 |
| [47] | Simpson, Carlos T., Moduli of representations of the fundamental group of a smooth projective variety. I, Publ. Math. IHÉS, 79, 47-129 (1994) · Zbl 0891.14005 |
| [48] | Simpson, Carlos T., Moduli of representations of the fundamental group of a smooth projective variety. II, Publ. Math. IHÉS, 80, 5-79 (1995), 1994 · Zbl 0891.14006 |
| [49] | Souto, Juan, A remark on the homotopy equivalence of \(SU_n\) and \(SL_n C (2010)\) |
| [50] | Steinberg, Robert, Regular elements of semisimple algebraic groups, Publ. Math. IHÉS, 25, 49-80 (1965) |
| [51] | Thurston, William P., Three-Dimensional Geometry and Topology, vol. 1, Princeton Mathematical Series, vol. 35 (1997), Princeton University Press: Princeton University Press Princeton, NJ, edited by Silvio Levy · Zbl 0873.57001 |
| [52] | Witten, Edward, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys., 5, 5, 841-907 (2001) · Zbl 1019.81040 |
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