On sharpness in local converse theorems for classical groups and G2

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 2 N ; 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 2 N .