Distinguished representations of \(\mathrm{GL}(n)\) and local converse theorems
Let \(E\) be a quadratic extension of a nonarchimedean local field \(F\) and \(\pi\) be an irreducible supercuspidal representation of \(\mathrm{GL}_n(E)\). Assume \(\psi\) is a nontrivial character of \(E\) whose restriction to \(F\) is trivial. The paper gives a criterion on when \(\pi\) is distinguished over \(\mathrm{GL}_n(F)\). The criterion is that the central character of \(\pi\) is trivial on \(F^\times\) and the Rankin-Selberg gamma factors \(\gamma(\frac12, \pi\times\tau,\psi)=1\) for all \(\mathrm{GL}_{n-2}(F)\) distinguished, irreducible unitary generic representation \(\tau\) of \(\mathrm{GL}_{n-2}(E)\). In the analogous situation where \(E=F\oplus F\), the above result gives a criterion on when two irreducible supercuspidal representations \(\pi_1,\pi_2\) are isomorphic, namely when the gamma factors \(\gamma(\frac12,\pi_i\times\tau,\psi)\) match. This is a (previously known) local converse theorem.
scientific article; zbMATH DE number 6477572
scientific article