Character variety - Imported Knowlpedia Content

Definition

Let $k$ be an algebraically closed field of characteristic zero, let $\Gamma$ be a finitely generated group, and let $G$ be a reductive affine algebraic group over $k$ . The $G$ -character variety of $\Gamma$ is the affine categorical quotient

X_G(\Gamma) = \operatorname{Hom}(\Gamma,G)\mathbin{/\mkern-6mu/}G := \operatorname{Spec}\!\left( k[\operatorname{Hom}(\Gamma,G)]^G \right),

where $G$ acts on the representation variety by conjugation. Its regular functions are exactly the conjugation-invariant regular functions on $\operatorname{Hom}(\Gamma,G)$ .

Closed-orbit interpretation

The quotient map is constant on conjugacy orbits, but it need not distinguish all of them. Each closed point of $X_G(\Gamma)$ corresponds to a unique closed orbit contained in an orbit closure. For $G=\operatorname{GL}_n$ , the closed orbits are the completely reducible representations. This geometric-invariant-theory interpretation is established in Sikora, §§7 and 11.

Relation to flat connections

When $\Gamma=\pi_1(M,x)$ , conjugation records the change of a framing in the fiber over $x$ for a holonomy representation. Thus character varieties provide algebraic models for moduli of flat connections. Their smooth loci can carry additional geometry; for surface groups and suitable $G$ , the good-representation locus has Goldman's symplectic structure Goldman, 1984.

Examples and scope

For $\Gamma=\mathbb Z$ , the representation variety is $G$ , and the character variety is the affine quotient $G\mathbin{/\mkern-6mu/}G$ by conjugation. For $G=\operatorname{GL}_n$ , invariant functions include traces of words in the images of generators.

References

Adam S. Sikora, “Character Varieties,” Transactions of the American Mathematical Society 364 (2012), 5173–5208. DOI record. Relevant: §§5, 7, and 11, representation varieties, closed orbits, and the categorical quotient.
William M. Goldman, “The Symplectic Nature of Fundamental Groups of Surfaces,” Advances in Mathematics 54 (1984), 200–225. DOI record. Relevant: the symplectic structure on surface-group representation moduli.