What I mean by classical: For the case of $GL\_2$, the answer to my question would be that the automorphic forms are either Maas forms or modular forms. For $GSp(2n)$ these are the Siegel modular forms.


Are there any "classical" objects which appeared in mathematics before this automorphic perspective which align with automorphic forms on $U(n)$ or other unitary groups? 


Maybe its the case that, historically, automorphic forms on unitary groups came first: then the question is: are there any other "models" of these kinds of functions?



"Classical" description of automorphic forms on unitary groups
These go by the name of "Hermitian modular forms". They occur very frequently in papers of Shimura (e.g. his monograph *Arithmeticity on the theory of automorphic forms*) and in other more recent works. For instance, [this paper by Bouganis](https://www.math.uni-bielefeld.de/documenta/vol-20/37.pdf) describes in detail at the beginning of section 2 the classical description of Hermitian modular forms for $U(n, n)$, the adelic description, and the passage between the two. There are similar descriptions available for $U(m, n)$ with $m \ne n$ as well.


