I know that the automorphic representation can be defined only for reductive algebraic group.


What property of algebraic group makes it hinder to define for all algebraic group and what nice property of reductive enable us define automorphic representation to it?


And I also wondering whether unitary group is reductive group. I think it does so.


But for me, it is not clear why they are reductive. It's hard to guess it just from the definition of reductiveness.


Would you shed me light on this?



On the reductive group
It is not at all that automorphic representations can be defined only for reductive algebraic groups, but, rather, that the essential difficulties arise in that case... as opposed to (for example) abelian unipotent (e.g., additive groups). Indeed, "Jacobi modular forms" are modular forms (in effect) for parabolic subgroups of reductive groups.


There is no obstacle to ask about irreducible subrepresentations (e.g.) of $L^2(G\_k\backslash G\_\mathbb A)$ or related things for general $G$, rather than merely reductive: the question makes sense. 


Unitary groups are indeed reductive, as can be verified by testing their (algebraic) Lie algebras. No, the definition of reductiveness itself is not directly helpful. The fact that the complexification of $U(p,q)$ is $GL(p+q,\mathbb C)$ is more-or-less a proof that $U(p,q)$ is reductive, since "we know that" $GL(p+q)$ is reductive.


Generally, the "definition"s of "reductive", "parabolic subgroup", and so on, are not so easy to check for the classical groups and the corresponding subgroups... Indeed. So if/when one finds this difficulty, it should not be surprising, but it is not evidence for any genuine issue, since the notions of "reductive", etc., were formed many decades after these classical (reductive or semi-simple) groups and their "parabolic" subgroups were very well understood.


