Consider a semisimple Lie group or a $p$ adic reductive group $G$.
To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
For a reductive Lie group, the character characterizes an irreducible admissible representation up to infinitesimal equivalence. Referring to Knapp's "Representation Theory, etc", Proposition 10.5 says that two infinitesimally-equivalent irreducible admissible representations have the same character, and Theorem 10.6 says that infinitesimally-inequivalent irreducible admissible representations have linearly independent characters.
For reductive $p$-adic groups, the character characterizes irreducible admissible representations., in that inequivalent irreducible admissible representations have linearly independent characters. See, e.g., Section 17 of Murnaghan's notes.