Restriction map between spaces of automorphic forms Hello,
Let $H \subset G$ be reductive groups defined over $\mathbb{Q}$. I consider the spaces of automorphic forms of $G$ and $H$. One has a restriction map from the space of automorphic forms of $G$ to the space of automorphic forms of $H$. Let us denote by $\rho$ this map.
I want to know in which directions $\rho$ has been studied. I am especially interested in the question of surjectivity.  More specifically, I consider the following situation:
Let $k_0$ be a totally real number field and $k$ a quadratic imaginary extension of $k_0$. Let $V$ be a $k$-vectorial space endowed with an anisotropic hermitian product (for the non-trivial Galois automorphism). Denote by $G$ the corresponding unitary group. It is a reductive group, defined over $k_0$ and compact at the archimedean places. The group $G$ contains a subgroup $H$ defined as the stabilizer of a decomposition $V = V_1 \oplus V_2$, where $V_i$ is one-dimensional. Thus, at an archimedean place, the inclusion $H \subset G$ is an inclusion $U(1) \times U(1) \subset U(2)$.
One uses Weil restriction to define these groups over the rational numbers and I want to know if the restriction map defined above is surjective.
Do you have any references for this kind of problem ?
Thanks.
 A: For semisimple algebraic groups, look at Proposition 1 in Akshay Venkatesh's article here:
http://math.stanford.edu/~akshay/research/bs.pdf
The results seems due to Clozel and Ullmo. The references point to a preprint of them which I couldn't track down.
For the special consideration indicated at the end, I have no clue whether things are surjective:( Certainly induction isn't, since cuspidal components for GL(n) at p-adic places can't be produced that way.
A: An automorphic form on $G$ apart from being $G({\mathbb Q})$ invariant, is annihilated by an ideal of finite codimension in the enveloping algebra of the lie algebra of $G({\mathbb R})$. So, it is not true that restriction of an automorphic form on $G$ to $H$ gives an automorphic form on $H$. Can you clarify what you had in mind?  
A: A belated data-point: in restricting a Siegel-type Eisenstein series from $Sp_2$ (meaning $4$-by-$4$ matrices) to the obvious imbedded copy of $SL_2\times SL_2$, any $f\otimes F$ for (strong-sense) cuspforms $f,F$ downstairs is orthogonal to the restriction. (When $f\sim F$, one obtains essentially the symmetric square $L$-function.) This is the content of my old "restrictions of Eisenstein series" (as in the Katata volume from 1984, and then also Inv. Math. 1993 treating the analogue for $Sp_n\times Sp_n\subset Sp_{2n}$), and also PiatetskiShapiro-Rallis "doubling method" from about the same time.
