# “Classical” description of automorphic forms on unitary groups

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?

• By their L-functions ? – reuns Nov 8 '16 at 0:08
• I have no idea if such a thing exists in the literature, but one way to go about it would be to start with the adèlic picture of automorphic representations of unitary groups and work backwards. For $\mathrm{GL}_2$, this isn't too hard to do once the dictionary between classical automorphic forms on the upper half plane and automorphic representations of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ is fleshed out (and this exists in the literature, more or less, in the books of Gelbart, Bump, and Goldfeld-Hundley). – Peter Humphries Nov 8 '16 at 0:15

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 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.
• Also, Hel Braun had looked at holomorphic automorphic forms on $U(n,n)$ in the 1950s in a style similar to what she and C. Siegel had used c. 1939 in studying holomorphic modular forms on $Sp(n,\mathbb R)$. – paul garrett Nov 8 '16 at 12:35