I have a question on how to define a trace of a unitary representation. If $G$ is a reductive group over a p-adic field it is known ( I do not know who prove it) that is of type I. Knowing this we have that every representation is traceable. However I think that there are more than one way to get a trace of a representation but only one canonical. For instance the regular representation has $f(1)$.

What will be the appropriate trace for the induced representation of the trivial representation of a subgroup $N$? i.e. what is correct trace for $c-ind_N^G1_n$?