Question

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 $c-ind_N^G1_n$?

Answer 1

If $G$ is a $p$-adic reductive group, it is a result of Jacquet that any irreducible smooth representation $(\pi ,V)$ of $G$ is admissible : for any compact open subgroup $K$ of $G$, the fixed vector space $V^K$ is finite dimensional. From this it follows that for all locally constant function $f$ on $G$ with compact support, the operator $\pi (f) \in {\rm End}(V)$ has finite range, and therefore has a trace.

If $N$ is not cocompact in $G$, the compactly induced representation ${\rm c-Ind}_N^G {\mathbf 1}$ is not admissible in general. Moreover it is not semisimple in general neither of finite length.

For more information you may read Bushnell's paper "Induced representations of locally profinite groups".

Answer 2

There is a notion of a CRR group in Dixmier's book on $C^*$ algebras, which is a group such that for every unitary irreducible representation $\pi$, the integrated representation $\int \pi$, a representation of $C_0(G)$, is trace class, if you restrict to compactly supported function.

The expression $tr \int \pi(f)$ is the trace of the representation here.

Lie groups, abelian locally compact groups, reductive p-adic groups all are CRR groups.

I do not have the book with me, but you should find it in the index of the mentioned book.

Now to your main question. I assume for simplicity that the subgroup $N$ is cocompact in $G$, then $c-ind$ and $Ind$ are isomorphic functors, then the chapter Selberg trace formula in Deitmar-Echterhoff "principles of harmonic analysis" proves this fact.

If you look at arbitrary subgroups, I am not sure, if $c-ind_N^G 1$ has a nice decomposition into irreducibles with finite multiplicity, but I am not an expert, so you might want to wait until somebody knows more.