Local Shalika models and functoriality (Q958179)

This paper relates local Langlands functorial lifts and Shalika models. Let \(F\) be a \(p\)-adic local field. The main result is the following: an irreducible supercuspidal representation of \(\text{GL}_{2n}(F)\) is a local Langlands functorial transfer from \(\text{SO}_{2n+1}(F)\) if and only if it has a nonzero Shalika model. Shalika models are defined for representations of \(\text{GL}_{2n}(F)\). In order to prove the main result, the authors consider generalized Shalika models defined for representations of \(\text{SO}_{4n}(F)\). They show that an irreducible admissible representation of \(\text{SO}_{4n}(F)\) cannot have both nonzero generalized Shalika model and a nonzero Whittaker model. This is used for proving that if an irreducible unitary supercuspidal representation \(\tau\) of \(\text{GL}_{2n}(F)\) has a nonzero Shalika model, then the unitarily induced representation \(I(s, \tau)\) of \(\text{SO}_{4n}(F)\) reduces at \(s=1\). This is then related to various known characterizations of the local Langlands transfer property. Altogether, they are stated as follows: Theorem. Let \(\tau\) be an irreducible supercuspidal representation of \(\text{GL}_{2n}(F)\). Then the following are equivalent. (1) \(\tau\) has a nonzero Shalika model. (2) The local exterior square \(L\)-factor \(L(s, \tau, \wedge^2)\) has a pole at \(s=0\). (3) The local exterior square \(\gamma\)-factor \(\gamma(s, \tau, \wedge^2, \psi)\) has a pole at \(s=1\). (4) The unitarily induced representation \(I(1,\tau)\) of \(\text{SO}_{4n}(F)\) is reducible. (5) \(\tau\) is a local Langlands functorial transfer from \(\text{SO}_{2n+1}(F)\). If one of the above holds for \(\tau\), then \(\tau\) is self-dual. As an application of their results, the authors prove three conjectures in the theory of automorphic forms.