Let $G$ be a connective reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definition of these terminology, please refer to I.1.4 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

Let $\mathbb{A}$ be a adele ring of $\mathbb{Q}$ and $\mathcal{A}_P(G)$ be a space of automorphic forms on $N(\mathbb{A})P(\mathbb{Q}) \backslash G(\mathbb{A})$. (For the definition of automorphic forms, see I.2.17 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

The constant map $\phi \to \phi_P$ from $\mathcal{A}_G(G)$ to $\mathcal{A}_P(G)$ is defined by $\phi_P(g)=\int_{N(\mathbb{Q}) \backslash N(\mathbb{A})}\phi(ng) dn$.

Then I am wondering whether that this map is surjective. Is this map surjective?

Any comments will be highly appreciated!

Let $G$ be a connected reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definitions of these terms please refer to I.1.4 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

Let $\mathbb{A}$ be a adele ring of $\mathbb{Q}$ and $\mathcal{A}_P(G)$ be a space of automorphic forms on $N(\mathbb{A})P(\mathbb{Q}) \backslash G(\mathbb{A})$. (For the definition of automorphic forms, see I.2.17 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

The constant map $\phi \to \phi_P$ from $\mathcal{A}_G(G)$ to $\mathcal{A}_P(G)$ is defined by $\phi_P(g)=\int_{N(\mathbb{Q}) \backslash N(\mathbb{A})}\phi(ng) dn$.

Then I am wondering whether that this map is surjective. Is this map surjective?

Any comments will be highly appreciated!

Let $G$ be a connective reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definition of these terminology, please refer to I.1.4 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

Let $\mathbb{A}$ be a adele ring of $\mathbb{Q}$ and $\mathcal{A}_P(G)$ be a space of automorphic forms on $N(\mathbb{A})P(\mathbb{Q}) \backslash G$. (For the definition of automorphic forms, see I.2.17 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

The constant map $\phi \to \phi_P$ from $\mathcal{A}_G(G)$ to $\mathcal{A}_P(G)$ is defined by $\phi_P(x)=\int_{N(\mathbb{Q}) \backslash N(\mathbb{A})}\phi(ng) dn$.

Then I am wondering whether that this map is surjective. Is this map surjective?

Any comments will be highly appreciated!

Let $G$ be a connective reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definition of these terminology, please refer to I.1.4 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

Let $\mathbb{A}$ be a adele ring of $\mathbb{Q}$ and $\mathcal{A}_P(G)$ be a space of automorphic forms on $N(\mathbb{A})P(\mathbb{Q}) \backslash G(\mathbb{A})$. (For the definition of automorphic forms, see I.2.17 of Moeglin and Waldspurge's book 'Spectral decomposition and Eisenstein series')

The constant map $\phi \to \phi_P$ from $\mathcal{A}_G(G)$ to $\mathcal{A}_P(G)$ is defined by $\phi_P(g)=\int_{N(\mathbb{Q}) \backslash N(\mathbb{A})}\phi(ng) dn$.

Then I am wondering whether that this map is surjective. Is this map surjective?

Any comments will be highly appreciated!