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.
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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 $d = \sum n_i d_i$.