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

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 The construction map is simply matter assembling arguments that are already in literature, together previously unpublished theorem G. Savin on exceptional theta correspondences, included as an appendix.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