Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups.
Let $G$, $H$ be two reductive groups defined over a number field $F$. Let $\mathcal{A}(G)$ be the space of automorphic forms on $G(F)\backslash G(\mathbb{A})$ and $L^{2}(G):=L^{2}(G(F)\backslash G(\mathbb{A})^{1})$ as in e.g. J. Arthur's papers . When are the following isomorphisms true (as representations)? $$ \mathcal{A}(G\times H)\cong\mathcal{A}(G)\otimes\mathcal{A}(H)$$ and $$ L^{2}(G\times H)\simeq L^{2}(G)\hat{\otimes}L^{2}(H)$$
How to obtain the spectral decomposition of $L^{2}(G\times H)$ (as direct integral of irreducibles) from the spectral decompositon of $L^{2}(G)$ and $L^{2}(H)$ ?
The examples I have in mind is $G=GL_{n}$ and $H=GL_{m}$ or $G=SO_{n}$ and $H=SO_{m}$. Are they true in these case?
Note that the isomorphisms here are both for representations! So the question may be not so trivial.
Thank you so much! Any comments, example or non-example will be welcomed!
