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Register a new userLocal Langlands Conjecture for $p$-adic $GSpin_4,$ $GSpin_6,$ and their inner forms
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We establish the local Langlands conjecture for small rank general spin groups $GSpin_4$ and $GSpin_6$ as well as their inner forms. We construct appropriate $L$-packets and prove that these $L$-packets satisfy the properties expected of them to the extent that the corresponding local factors are available. We are also able to determine the exact sizes of the $L$-packets in many cases.
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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of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix.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 $G_2$.
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