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.