Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $c-ind_N^G1_n$?

share|cite|improve this question

2 Answers 2

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".

share|cite|improve this answer
+1 Nice reference. – Marc Palm Jun 24 '11 at 11:17

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.

share|cite|improve this answer
Lie groups are not CCR, only reductive ones are. – Marc Palm Mar 5 '13 at 6:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.