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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 $SU(2)$ and some corresponding character varieties, but somehow these invariants can "see" representations into $SL(2,\mathbb C)$ as well (in particular, the volumes of these representations). Thus it is natural to ask how the theory would/could extend to $SL(2,\mathbb C)$.

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 $X:=\operatorname{Hom}(\pi_1(\Sigma),SU(2))/\\!/SU(2)$. 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 $\mathcal L$ ($\mathcal L$ 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 $SU(2)$ is compact). But what if I want to work with the $SL(2,\mathbb C)$ character variety instead? Let's call this character variety $X_{\mathbb C}$ (note that everything I've said above about the symplectic form, Kahler 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 $SU(2)$ and the complex points are $SL(2,\mathbb C)$. Using these coordinates for the construction of character varieties shows that $X_{\mathbb C}$ is indeed the complexification of $X$.

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Once again, the SU(2) characters are not all of the real points of the SL(2,C) character variety. –  Charlie Frohman Jul 30 '11 at 21:22
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