In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split G 2 over finite fields k of odd characteristic. Then we prove the first converse theorem for exceptional groups, namely, GL 1 and GL 2 -twisted gamma factors will uniquely determine an irreducible generic cuspidal representation of G 2 (k).
√q + 1. In future work, the authors plan to check this expectation directly by analyzing GL 1 -twisted gamma factors for irreducible generic cuspidal representations of G 2 (F q ). Theorem 1.3 inspires us to consider the local converse problem for G 2 (F ) when F is a p-adic field. In this case, our proof in §6 is actually valid for an analogue of Proposition 1.5 without the restriction that Π is cuspidal, which gives us the local functional equation of the local zeta integral of Piatetski-Shapiro-Rallis-Schiffmann ([PSRS92]) and hence the existence of the GL 2 -twisted local gamma factors. However, the existence of the GL 1 -twisted local gamma factors relies on the following Conjecture 1.6. Let F be a p-adic field and Π be an irreducible generic representation of G 2 (F ). Let ψ be a nontrivial additive character of F . Let I(χ, ψ) be the genuine induced representation on the double cover SL 2 (F ) for a character χ of F × . Then if I(χ, ψ) is irreducible, we haveNote that both I(χ, ψ) and ω ψ are genuine representations on a double cover of J and the thus the tensor product I(χ, ψ) is a representation on J.In the above conjecture, we keep the requirement minimal so that it is enough to deduce the local functional equation of Ginzburg's local zeta integral ([Gi93]). We do expect that the following generalized conjecture is true Conjecture 1.7. Let F be a p-adic field and Π be an irreducible (selfdual) representation of G 2 (F ). Let ψ be a nontrivial additive character of F . Let π be an irreducible genuine representation on the double cover SL 2 (F ). Then we haveAs explained in [LZ18, §6], Conjecture 1.7 is an analogue of the uniqueness problem of Fourier-Jacobi models for Sp 2n , which was proved in [BR00] (for n = 2) and in [GGP12, Su12](for general n). Once Conjecture 1.6 is established, we then have the local gamma factors for irreducible generic representations of G 2 (F ) × GL 1 (F ) using Ginzburg's local zeta integral. Inspired by Theorem 1.3, we propose the following conjecture on the local converse problem for G 2 (F ).Conjecture 1.8. Let F be a p-adic field. Suppose that Conjecture 1.6 is true. Let Π 1 , Π 2 be two irreducible generic representations of G 2 (F ). Iffor all characters χ of GL 1 (F ) and all irreducible generic representations τ of GL 2 (F ), then Π 1 ∼ = Π 2 .Conjectures 1.6-1.8 are current work in progress of the authors. Again, by Langlands philosophy of functoriality, representations of G 2 (F ) are expected to be lifted to representations of GL 7 (F ) and this lifting is expected to preserve GL-twisted local gamma factors. Then the Jacquet's local converse conjecture for GL n , which was recently...
