I'm trying to understand a few things about automorphic L-functions. In page 5 of <http://arxiv.org/pdf/1401.0390.pdf>, the author mentions the isobaric sum decomposition $\pi=n\_{1}\pi\_{1}\boxplus\cdots\boxplus n\_{r}\pi\_{r}$. Let $d(\pi)$ denote the dimension of the representation space of $\pi$. Does the following equality hold true?
$d(\pi)=\displaystyle{\sum\_{i=1}^{r}n\_{i}d(\pi\_{i})}$  

Thanks in advance.



degree of an isobaric sum
No, it has nothing to do with the dimension of the representation space. (Automorphic representations of $GL\_n$ are always infinite-dimensional for $n > 1$.) The isobaric sum operation **changes the group**: each $\pi\_i$ is an automorphic representation of $GL(d\_i)$ for some $d\_i$, and the isobaric sum is an automorphic representation of $GL(d)$ where $d = \sum n\_i d\_i$. 


