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?
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 infinitesimallyequivalent irreducible admissible representations have the same character, and Theorem 10.6 says that infinitesimallyinequivalent 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. 

