TY - JOUR
T1 - A general Cayley correspondence and higher rank Teichmuller spaces
AU - Bradlow, Steven
AU - Collier, Brian
AU - Garc__a-Prada, Oscar
AU - Gothen, Peter B.
AU - Oliveira, Andr_e
N1 - Publisher Copyright:
© 2024 Department of Mathematics, Princeton University.
PY - 2024
Y1 - 2024
N2 - We introduce a new class of sl2-triples in a complex simple Lie algebra g,which we call magical. Such an sl2-triple canonically de_nes a real form and.various decompositions of g. Using this decomposition data, we explicitly.parametrize special connected components of the moduli space of Higgs.bundles on a compact Riemann surface X for an associated real Lie group,hence also of the corresponding character variety of representations of _1X.in the associated real Lie group. This recovers known components when.the real group is split, Hermitian of tube type, or SOp;q with 1 < p 6 q,and also constructs previously unknown components for the quaternionic.real forms of E6, E7, E8 and F4. The classi_cation of magical sl2-triples is.shown to be in bijection with the set of _-positive structures in the sense.of Guichard{Wienhard, thus the mentioned parametrization conjecturally.detects all examples of higher rank Teichmuller spaces. Indeed, we discuss.properties of the surface group representations obtained from these Higgs.bundle components and their relation to _-positive Anosov representations,which indicate that this conjecture holds.
AB - We introduce a new class of sl2-triples in a complex simple Lie algebra g,which we call magical. Such an sl2-triple canonically de_nes a real form and.various decompositions of g. Using this decomposition data, we explicitly.parametrize special connected components of the moduli space of Higgs.bundles on a compact Riemann surface X for an associated real Lie group,hence also of the corresponding character variety of representations of _1X.in the associated real Lie group. This recovers known components when.the real group is split, Hermitian of tube type, or SOp;q with 1 < p 6 q,and also constructs previously unknown components for the quaternionic.real forms of E6, E7, E8 and F4. The classi_cation of magical sl2-triples is.shown to be in bijection with the set of _-positive structures in the sense.of Guichard{Wienhard, thus the mentioned parametrization conjecturally.detects all examples of higher rank Teichmuller spaces. Indeed, we discuss.properties of the surface group representations obtained from these Higgs.bundle components and their relation to _-positive Anosov representations,which indicate that this conjecture holds.
KW - character varieties
KW - Higgs bundles
KW - higher Teichmuller spaces
KW - magical sl2-triples
KW - representations of surface groups
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U2 - 10.4007/annals.2024.200.3.1
DO - 10.4007/annals.2024.200.3.1
M3 - Article
AN - SCOPUS:85203398968
SN - 0003-486X
VL - 200
SP - 803
EP - 892
JO - Annals of Mathematics
JF - Annals of Mathematics
IS - 3
ER -