Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $\rho$ of the skein algebra, which is a character $r_\rho \in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ represented by a group homomorphism $\pi_1(S) \to \mathrm{SL}_2(\mathbb C)$. The main result of the current article is that, when the surface $S$ is closed, every character $r\in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism $r\colon \pi_1(S) \to \mathrm{SL}_2(\mathbb C)$ a representation of the skein algebra $\mathcal S^A(S)$ that is uniquely determined up to isomorphism.