I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it! 


The question is simple: let $G$ be a reductive (or even semisimple) algebraic group over $\mathbb Q$. Is it true that the adelic group $G(\mathbb A)$ is of Type I (i.e., direct integral decomposition of its unitary representations is unique)? And if the answer is negative, then how do people actually get around it in the study of multiplicity of automorphic forms in $L^2(G(\mathbb Q)\backslash G(\mathbb A))$?



Is a reductive adelic group a Type I group?
I believe the answer is yes. Let's begin by recalling that if one wants to show that a locally compact group $G$ is of type I, it suffices to show that $G$ contains a "large" compact subgroup $K$, in the sense that for every $\pi \in \hat{G}$ and $\sigma \in \hat{K}$, the multiplicity of $\sigma$ in $\pi|\_K$ is finite. This is how Harish-Chandra showed that a real reductive group is of type I (take $K$ to be a maximal compact), and also how [Bernstein](http://www.math1.tau.ac.il/~bernstei/Publication_list/publication_texts/bernstein-P-tame-FAN.pdf) showed that a $p$-adic reductive group is of type I (take $K$ to be a compact open subgroup).


Now let $G$ be a connected reductive group over $\mathbb Q$. Then, away from a finite set $S$ of places (containing $\infty$), $G$ is unramified and has a model over $\mathbb Z\_p$. Let's abuse notation and denote this model by $G$. It suffices to show that $G(\mathbb A^S) = \prod'\_{p \not\in S} G(\mathbb Q\_p)$ is of type I. The desired large $K$ turns out to be $K = \prod\_{p \not\in S} G(\mathbb Z\_p)$. This assertion essentially appears (without proof) as Theorem 4 in Flath's article in the Corvallis proceedings. The details are spelled out in the appendix to Clozel's article in the IAS/Park City 2002 lecture notes on automorphic forms ([MR2331351](http://www.ams.org/mathscinet-getitem?mr=2331351); a Google Books preview is available [here](http://books.google.ca/books?id=EYsKo1FrLfIC&lpg=PA41&ots=1mwlueepxZ&pg=PA87#v=onepage&q&f=false)).


