On sharpness in Local Converse Theorems for classical groups and $G_2$

Abstract: We prove various results about the Local Converse Problem for split reductive groups $G$ over a non-archimedean local field~$F$ of characteristic $0$ and residual characteristic $p$. In particular, we prove that when $G$ is a symplectic or special orthogonal group, or the exceptional group $G_2$, and $p$ is large enough, then the optimal standard Local Converse Theorem for $G(F)$ requires twisting by representations of $GL_r(F)$ with $r$ up to half the dimension of the standard representation of the dual group of $G$. However, if we restrict to generic supercuspidal representations of $G(F)$ then it can be improved when $G=SO_{2N}$; we conjecture that the same is true for symplectic and odd special orthogonal groups. We also consider the possibility of using non-standard representations of the dual group to distinguish representations, giving counterexamples to possible improvements for general linear groups, $G_2$ and $SO_{2N}$.