Higgs bundles and surface group representations in the real symplectic group

Garcı́a-Prada, Oscar; Gothen, Peter B.; Riera, Ignasi Mundet i

doi:10.1112/jtopol/jts030

Cited by 57 publications

(96 citation statements)

References 48 publications

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“…For an arbitrary real reductive Lie group G, the definition of stability and semistability is significantly more involved, see [11,13]. However, if G ⊂ SL(n, C) then a G-Higgs bundle is unstable if and only if the corresponding SL(n, C)-Higgs bundle is unstable.…”

Section: Remark 23mentioning

confidence: 99%

Maximal $${\mathsf {Sp}}(4,{\mathbb {R}})$$ surface group representations, minimal immersions and cyclic surfaces

Collier

2015

Geom Dedicata

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Let S be a closed surface of genus at least 2. For each maximal representation ρ : π 1 (S)→Sp(4, R) in one of the 2g − 3 exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space Sp(4, R)/U(2) is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmüller space. Unlike Labourie's recent results on Hitchin components, these bundles are not vector bundles.

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Section: Remark 23mentioning

confidence: 99%

Maximal $${\mathsf {Sp}}(4,{\mathbb {R}})$$ surface group representations, minimal immersions and cyclic surfaces

Collier

2015

Geom Dedicata

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“…When the group

G

is a classical group, or more generally when

H

is a classical group, it is useful to take the standard representation of

H^{C}

to describe a

G

‐Higgs bundle in terms of associated vector bundles. This is the approach taken in .…”

Section: Hermitian Groups Toledo Invariant and Milnor–wood Inequalitymentioning

confidence: 99%

“…Theorem was proved on a case by case basis for the classical groups . In these references, the bound given is for the integer

d \in π_{1} false(H false) ≅ π_{1} false(H^{C} false) ≅ Z

associated naturally to the

H^{C}

‐bundle

E

.…”

Section: Hermitian Groups Toledo Invariant and Milnor–wood Inequalitymentioning

confidence: 99%

“…As mentioned above, our results should provide the starting point for an intrinsic study of the topology and geometry of the moduli spaces of

G

‐Higgs bundles when

G

is a group of Hermitian type, in particular for the counting of connected components of the moduli space. This had been carried out to some extent for some of the classical groups on a case by case basis , making use of the classification theorem of Lie groups, but no general principle emerged. Our present intrinsic approach offers a new understanding of the Toledo invariant, Milnor–Wood bound and rigidity phenomena, which are fundamental to the topological study of the moduli space.…”

Section: Introductionmentioning

confidence: 99%

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Higgs bundles, the Toledo invariant and the Cayley correspondence

Biquard

Garcı́a-Prada

Rubio

2017

Journal of Topology

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Abstract. Motivated by the study of the topology of the character variety for a non-compact Lie group of Hermitian type G, we undertake a uniform approach, independent of classification theory of Lie groups, to the study of the moduli space of G-Higgs bundles over a compact Riemann surface. We give an intrinsic definition of the Toledo invariant of a G-Higgs bundle which relies on the Jordan algebra structure of the isotropy representation for groups defining a symmetric space of tube type, and prove a general Milnor-Wood type bound of this invariant when the G-Higgs bundle is semistable. Finally, we prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when G is of tube type, which reveals new topological invariants only seen in particular cases from the character variety viewpoint.

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“…As far as we know, Theorem 5.13 has not been proved in general for the case of G-Higgs bundles with G not connected (see however [8,20] for definitions and some non-connected real reductive groups). On the other hand, our results seem to indicate that this form of the non-abelian Hodge theorem should still be valid in this case, for an adequate notion of stability of G-Higgs bundles, defined in terms of reduction of structure group to R-parabolics.…”

Section: Stability In Representation Varietiesmentioning

confidence: 99%

Stability of Affine G-Varieties and Irreducibility in Reductive Groups

Florentino

Casimiro

2012

Int. J. Math.

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Let G be a reductive affine algebraic group and let X be an affine algebraic G-variety. We establish a (poly)stability criterion for points x ∈ X in terms of intrinsically defined closed subgroups Hx of G and relate it with the numerical criterion of Mumford and with Richardson and Bate-Martin-Röhrle criteria, in the case X = G N . Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions. the image of the representation, x(Γ) ⊂ G, is irreducible or completely reducible as a subgroup of G.In this paper, we generalize these relationships to a bigger class of affine G-varieties, where G is an affine reductive group, not necessarily irreducible, defined over an algebraically closed field k, of characteristic zero.Besides its intrinsic relevance as stability criteria, the constructions examined here perfectly agree with other known results for certain specific classes of G-varieties. For example, the non-abelian Hodge theorem (see [9] or [32]), in the case of complex reductive groups, can be restated as a correspondence between stable points in distinct (however homeomorphic) varieties. Also, from our set up, one can recover the Mundet-Schmitt criterion for polystability [18].To describe our main results, let Y (G) denote the set of one parameter subgroups (1PS for short) of G, that is, homomorphisms λ from the multiplicative group k × to G. Given a 1PS λ ∈ Y (G) and g ∈ G, the morphism t → λ(t)gλ(t) −1 may or may not extend to k (as a morphism of affine varieties). In case there is such an extension, we say that lim t→0 λ(t)gλ(t) −1 exists. It is known that P (λ) = g ∈ G : lim t→0 λ(t)gλ(t) −1

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Higgs bundles and surface group representations in the real symplectic group

Cited by 57 publications

References 48 publications

Maximal $${\mathsf {Sp}}(4,{\mathbb {R}})$$ surface group representations, minimal immersions and cyclic surfaces

Maximal $${\mathsf {Sp}}(4,{\mathbb {R}})$$ surface group representations, minimal immersions and cyclic surfaces

Higgs bundles, the Toledo invariant and the Cayley correspondence

Stability of Affine G-Varieties and Irreducibility in Reductive Groups

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