A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection $$\CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K)$$ from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $\rho : W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.