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Transactions of the American Mathematical Society

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Character varieties

Author: Adam S. Sikora
Journal: Trans. Amer. Math. Soc. 364 (2012), 5173-5208
MSC (2010): Primary 14D20; Secondary 14L24, 57M27
Published electronically: May 15, 2012
MathSciNet review: 2931326
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Abstract: We study properties of irreducible and completely reducible representations of finitely generated groups $ \Gamma $ into reductive algebraic groups $ G.$ In particular, we study the geometric invariant theory of the $ G$ action on the space of $ G$-representations of $ \Gamma $ by conjugation.

Let $ X_G(\Gamma )$ be the $ G$-character variety of $ \Gamma .$ We prove that for every completely reducible, scheme smooth $ \rho :\Gamma \to G$

$\displaystyle T_{[\rho ]}\, X_G(\Gamma )\simeq T_0\,\big (H^1(\Gamma ,Ad\,\rho )//S_\Gamma \big ),$

where $ H^1(\Gamma ,Ad\,\rho )$ is the first cohomology group of $ \Gamma $ with coefficients in the Lie algebra $ \mathfrak{g}$ of $ G$ twisted by $ \Gamma \stackrel {\rho }{\longrightarrow } G\stackrel {Ad}{\longrightarrow } GL(\mathfrak{g})$ and $ S_\Gamma $ is the centralizer of $ \rho (\Gamma )$ in $ G.$ The condition of $ \rho $ being scheme smooth is very important as there are groups $ \Gamma $ such that

$\displaystyle dim\, T_{[\rho ]}\, X_G(\Gamma )< T_0\, H^1(\Gamma ,Ad\, \rho ),$

for a Zariski open subset of points in $ X_G(\Gamma ).$ We prove, however, that all irreducible representations of surface groups are scheme smooth.

Let $ M$ be an orientable $ 3$-manifold with a connected boundary $ F$ of genus $ g\geq 2.$ Let $ X_G^g(F)$ be the subset of the $ G$-character variety of $ \pi _1(F)$ composed of conjugacy classes of good representations $ \rho : \Gamma \to G,$ i.e., irreducible representations such that the centralizer of $ \rho (\Gamma )$ is the center of $ G.$ By a theorem of Goldman, $ X_G^g(F)$ is a holomorphic symplectic manifold. We prove that the set of good $ G$-representations of $ \pi _1(F)$ which extend to representations of $ \pi _1(M)$ is a complex isotropic subspace of $ X_G^g(F).$ It is Lagrangian, if these representations correspond to reduced points of the $ G$-character variety of $ M$. It is an open problem whether it is always the case.

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Additional Information

Adam S. Sikora
Affiliation: Department of Mathematics, 244 Math. Bldg., University at Buffalo, SUNY, Buffalo, New York 14260

Keywords: Representation variety, character variety, irreducible representation, completely reducible representation, Goldman symplectic form, $3$-manifold, Lagrangian submanifold
Received by editor(s): January 31, 2010
Received by editor(s) in revised form: May 25, 2010, and August 23, 2010
Published electronically: May 15, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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