How to interpret sections over the $\mathrm{SU}(2)$ character variety as sections over the $\mathrm{SL}(2,\mathbb{C})$ character variety?

Question

The motivation for this question comes from the Volume Conjecture of Kashaev-Murakami-Murakami. The Jones family of invariants of knots and $3$-manifolds can all be defined using $\mathrm{SU}(2)$ and some corresponding character varieties, but somehow these invariants can "see" representations into $\mathrm{SL}(2,\mathbb C)$ as well (in particular, the volumes of these representations). Thus it is natural to ask how the theory would/could extend to $\mathrm{SL}(2,\mathbb C)$.

In the Witten-Reshetikhin-Turaev TQFT, we associate to a surface $\Sigma$ a certain space $Z(\Sigma)$ as follows. There is a natural line bundle $\mathcal L$ over the character variety $$X:=\operatorname{Hom}(\pi_1(\Sigma),\mathrm{SU}(2))/ \mathrm{SU}(2).$$ There is a natural symplectic form on $X$, and choosing a complex structure on $\Sigma$ equips $X$ with a complex structure which together with the symplectic form makes $X$ a Kahler manifold. Then $Z(\Sigma)$ is the Hilbert space of square integrable holomorphic sections of $\mathcal L$ (note $\mathcal L$ carries a natural inner product, and the curvature form of the induced connection coincides with the natural symplectic form on $X$).

Everything works well here in part because $X$ is compact (trivially, since $\mathrm{SU}(2)$ is compact). But what if I want to work with the $\mathrm{SL}(2,\mathbb C)$ character variety instead? Let's call this character variety $X_{\mathbb C}$ (note that everything I've said above about the symplectic form, Kähler structure, and line bundle all extend naturally to $X_{\mathbb C}$).

Do the sections forming $Z(\Sigma)$ naturally extend to sections of $\mathcal L$ over $X_{\mathbb C}$? In what sense are they still square-integrable? In particular, $X_{\mathbb C}$ is now non-compact, so how do I pick a finite-dimensional subspace of the space of sections of $\mathcal L$?

A justification of the notation $X_{\mathbb C}$ is as follows. If we consider a point on the variety $x^2+y^2+z^2+w^2=1$ as representing the matrix $\left(\begin{smallmatrix}x+iy&z+iw\cr-z+iw&x-iy\end{smallmatrix}\right)$, then the real points are exactly $\mathrm{SU}(2)$ and the complex points are $\mathrm{SL}(2,\mathbb C)$. Using these coordinates for the construction of character varieties shows that $X_{\mathbb C}$ is indeed the complexification of $X$.

Answer 1

Let $K$ be a connected compact connected Lie group and let $G$ be the complexification of $K$. Then $G/K$ is an affine space. For any finitely presented $\Gamma$, $\mathrm{Hom}(\Gamma, K)\subset \mathrm{Hom}(\Gamma,G)$ and less obviously we have an inclusion of character varieties: $$\mathfrak{X}_{\Gamma}(K):=\mathrm{Hom}(\Gamma,K)/K\hookrightarrow \mathfrak{X}_{\Gamma}(G):=\mathrm{Hom}(\Gamma,G)/\!\!/G.$$

The polar decomposition writes $G=K\times \mathfrak{p}$ as a space (not a group). We therefore have a projection map: $G\to K$. This induces a projection mapping $\mathrm{Hom}(F_r,G)\to \mathrm{Hom}(F_r,K)$ for a rank $r$ free group $F_r$. The polar decomposition is $K$-equivariant, and thus this projection mapping descends to a mapping $\mathrm{Hom}(F_r,G)/K\to \mathrm{Hom}(F_r,K)/K$ and thus by inclusion (and restriction) to a projection $\mathrm{Hom}(\Gamma,G)/K\to \mathrm{Hom}(\Gamma,K)/K$ for any surjection $F_r\to \Gamma$.

Since $\mathrm{Hom}(\Gamma,G)/K$ projects to $\mathfrak{X}_{\Gamma}(G)$, we have a commutative diagram:

$$\begin{array}{ccc} \mathrm{Hom}(\Gamma,G)/K&\twoheadrightarrow& \mathfrak{X}_{\Gamma}(K)\\ \downarrow& &\downarrow\\ \mathfrak{X}_{\Gamma}(G)& =&\mathfrak{X}_{\Gamma}(G) \end{array} $$

Perhaps analyzing the fibers of the top and left surjections, and how they relate in $\mathrm{Hom}(\Gamma,G)/K$ would be helpful?