character variety

Motivations

Relation to stable vector bundles, Teichmüller space. attachments/Pasted%20image%2020220408092618.png # Definitions

Setup: $G\in \mathsf{Alg} {\mathsf{Grp}}_{/ {{\mathbf{C}}}} $ a reductive algebraic group and $\pi$ a finitely generated group.
Defining the representation scheme:
- Define \begin{align*}{\mathfrak{X}}(\pi, G) \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Grp}}(\pi, G)\end{align*} as a set, the representation scheme.
- This admits the stricture of a scheme, and in fact an affine algebraic variety, induced by the structure of $G$: since it is finitely generated, take a presentation $\mathsf{Free}_m \to \mathsf{Free}_n \to G\to 0$, so $G \cong \left\langle{\gamma_1,\cdots, \gamma_n \mathrel{\Big|}R_1,\cdots R_m}\right\rangle$, and embed ${\mathfrak{X}}(\pi, G) \hookrightarrow G{ {}^{ \scriptscriptstyle\times^{n} } }$ by $\rho \mapsto \rho(\gamma_1) \times \cdots \times \rho(\gamma_n)$.
There is an action $G \curvearrowright{\mathfrak{X}}(\pi, G)$ where $g\cdot \rho({-}) \coloneqq g\rho({-})g^{-1}$, which is the action of an affine algebraic group on an algebraic set. “Taking orbits” via a stack-theoretic quotient yields the character stack
\begin{align*}\widetilde{{\mathcal{M}}}(\pi, G) \coloneqq{\mathfrak{X}}(\pi, G)/G\end{align*}
- As a stack, this is the functor ${\mathsf{Aff}}{\mathsf{Sch}}\to {\mathsf{Grpd}}$ where $\operatorname{Spec}R \mapsto {}_{R\otimes{\mathbf{C}}[\pi_1 \Sigma_g]}{\mathsf{Mod}} $ which are locally free and rank $n$ over $R$.
Comparing the character stack to the GIT (categorical) quotient:
- Note: this quotient is the usual definition of character variety.
- Let ${\mathcal{O}}({\mathfrak{X}}(\pi, G))$ be the regular functions on the representation scheme and let ${\mathcal{O}}({\mathfrak{X}}(\pi, G))^G$ be its $G{\hbox{-}}$invariant functions under the conjugation action above.
- When it exists, define GIT character stack \begin{align*}{\mathcal{M}}(\pi, G) \coloneqq{\mathfrak{X}}(\pi, G) { \mathbin{/\mkern-6mu/}}G \coloneqq\operatorname{Spec}{\mathcal{O}}\qty{{\mathfrak{X}}(\pi, G)}^G,\end{align*}
- A sufficient condition for existence: $G$ being reductive.
- General fact: if $X$ is finite type then $X/G$ is again finite type?
- Finitely generated because ${\mathfrak{X}}(\pi, G)$ is affine and $G$ is reductive.
- Often called a variety in the literature, but is generally not irreducible.
- There is a map $\widehat{{\mathcal{M}}}(\pi, G) \to {{\mathcal{M}}}(\pi, G)$ which is a homeomorphism over the locus of irreps, and otherwise has fibers of reducible reps with equivalent semisimplification.
The term character variety is an eponym:
- For $g\in \pi$, there is an associated character $\chi_g: {\mathfrak{X}}(\pi, G) \to {\mathbf{C}}$ where $\rho \mapsto \operatorname{tr}^2 \rho(g)$.
- Define the character scheme as
  \begin{align*}\chi(\pi, G) \coloneqq\operatorname{Spec}{\mathbf{C}} { \left[ \scriptstyle {\left\{{\chi_g {~\mathrel{\Big\vert}~}g\in G}\right\}} \right] } .\end{align*}
  - For $G ={\operatorname{SL}}_n, {\mathsf{Sp}}_{2n}, {\operatorname{SO}}_{2n+1}$, and ${\operatorname{PSL}}_2({\mathbf{C}})$, ¹ \begin{align*}{\mathcal{M}}(\pi, G) \cong \chi(\pi, G).\end{align*}
  - Why this is true for ${\operatorname{PSL}}_2({\mathbf{C}})$: the invariant regular functions ${\mathcal{O}}({\mathfrak{X}}(\pi, G))^G$ is finitely generated since ${\mathfrak{X}}(\pi, G)$ is affine and $G$ is reductive, and is in fact generated by finitely many characters of the form $\chi_g(\rho) \coloneqq{\mathrm{tr}}^2\rho(g)$. Thus ${\mathcal{M}}(\pi, G)$ is a moduli space of ${\mathbf{C}}{\hbox{-}}$valued characters of $\pi$ factoring through $G$.

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The tangent spaces

For $\pi$ a surface group, the $G{\hbox{-}}$invariant locus $\chi_G^{\mathrm{irr}}(\pi)$ of irreducible reps with closed $G{\hbox{-}}$orbits is a complex manifold with manifold tangent spaces ${\mathbf{T}}_\rho(\chi_G^{\mathrm{irr}}(\pi)) \cong H^1_{\mathsf{Grp}}(\pi, {\mathfrak{g}}_{{ \operatorname{Ad} }\rho})$.
- Given a nondegenerate bilinear form $B:{\mathfrak{g}}{ {}^{ \scriptstyle\otimes_{{\mathbf{C}}}^{2} } }$, Goldman constructs a symplectic form $\omega = \omega(B)$ making $(\chi_G^{\mathrm{irr}}, \omega)$ a symplectic manifold.
Can associate a larger scheme $\widehat{\mathop{\mathrm{Hom}}}(\pi, G)$ with a larger structure sheaf whose closed points recover $\mathop{\mathrm{Hom}}_{\mathsf{Grp}}(\pi, G)$ as a scheme.
- The Zariski tangent spaces are isomorphic to 1-cocycles in group cohomology: ${\mathbf{T}}_\rho \widehat{\mathop{\mathrm{Hom}}}_{\mathsf{Grp}}(\pi, G) \cong Z^1_{\mathsf{Grp}}(\pi, {\mathfrak{g}}_{{ \operatorname{Ad} }\rho})$.

Yields a complex orbifold: attachments/Pasted%20image%2020220407202143.png

Examples

For $\Sigma_g$ a Riemann surface, there is an embedding of the Fricke space of marked hyperbolic structures $F(\Sigma_g) \hookrightarrow{\mathfrak{X}}(\pi_1\Sigma_g, {\operatorname{PSL}}_2({\mathbf{R}}))$ (note that $F(\Sigma_g) \cong {\mathcal{T}}(\Sigma_g)$, the Teichmüller space of complex structures on $\Sigma_g$)
- This is induced by the fact that a marked hyperbolic surface has a marking $\phi: \Sigma_g \to {\mathbb{H}}/\Gamma$ inducing $\pi_1\Sigma_g \to \Gamma\hookrightarrow\mathop{\mathrm{Isom}}^+({\mathbb{H}}) { \, \xrightarrow{\sim}\, }{\operatorname{PSL}}_2({\mathbf{R}})$.
- The image is the connected component of faithful and discrete reps.
Of character stacks:
- ${\mathcal{M}}(\pi_1 \Sigma_g, \operatorname{GL}_n({\mathbf{C}})) = {\mathcal{M}}^{\operatorname{Betti}}_{g, n}$ is a Betti moduli space appearing the in nonabelian Hodge correspondence. Concretely, \begin{align*}{\mathcal{M}}_{g, n}^{\operatorname{Betti}}= \left\{{ \left\{{A_i, B_i}\right\}_{1\leq i \leq g} {~\mathrel{\Big\vert}~}\prod_{1\leq i \leq g}[A_i, B_i] = \operatorname{id}}\right\}/\operatorname{GL}_n({\mathbf{C}}),\end{align*} the moduli stack of representations of $\pi_1 \Sigma_G$.
- ${\mathcal{M}}^{{\mathrm{ss}}}(\pi_1 \Sigma_g, {\operatorname{U}}_n)$ parameterizes semisimple reps, and polystable holomorphic bundles of degree zero on $\Sigma_g$.
- ${\mathcal{M}}(\mathsf{Free}_2, {\operatorname{SO}}_2) \cong S^1\times S^1$, a torus

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Notes

For $\pi = \pi_1\Sigma_g$ a surface group, the irreducible locus ${\mathcal{X}}^{\mathrm{irr}}(\pi, G)$ is a complex orbifold (Sik09) and Goldman computes $\dim_{\mathbf{C}}{\mathcal{X}}^{\mathrm{irr}}(\pi, G) = (2g-2)\dim_{\mathbf{C}}G + 2\dim_{\mathbf{C}}Z(G)$ where $Z(G)$ is the center of $G$.
For $\pi$ a surface group, the smooth locus consists of irreducible reps.