Suppose we have an automorphic form $f$ on $\Gamma\subset GL_n$ and certain homomorphism $\phi: GL_m\rightarrow GL_n$, then we know that the pullback of $f$ via $\phi$ is invariant under actions of $\phi^{-1}(\Gamma)$.
Is there anything (L-functions, spectral decomposition,...) we can say about this pullback? More conditions may be added when necessary. (Say, you may want that $f$ is some Eisenstein series, or $m=2$, or $\phi$ has some nice properties)
Thank you.
In some interesting cases, decomposing an automorphic form/representation on a bigger group $G$ along a smaller group $H$ produces Euler products and similar helpful outcomes. Perhaps not "in general", especially if the subgroup $H$ is "too small". Both Rankin-Selberg and Langlands-Shahidi -type presentations of $L$-functions can be viewed exactly as saying that the decomposition coefficients in such certain such situation are $L$-functions.
E.g., even in the very simplest circumstance, restricting an elliptic modular cuspform to the split torus and decomposing, the "decomposition coefficients" are the $L$-functions associated to the cuspform via Mellin transform.
The case $GL_{n-1}\subset GL_n$, spectrally decomposing a cuspform $f$ on the larger along the smaller, produces the more general analogue of the previous, namely, the Hecke-type Rankin-Selberg product $L(s,f\times F)$ where $F$ is a cuspform on the smaller.
In some cases, the decomposition of the restriction can be completely understood, as in the case of $Sp_m\times Sp_n \to Sp_{m+n}$ and restriction of Siegel-type Eisenstein series (e.g., see my Taniguchi/Katata Symposium paper, 1984).
The cuspidal components of the restriction to $SL_2\times SL_2\times SL_2\to Sp_3$ (or $Sp_6$, if one prefers) are triple-product $L$-functions.
But, yes, in general, one cannot expect to say much.
I don't think you can say many useful things. E.g., if you conider an embedding of a Levi $M \subset GL(2)$, then the pullback should correspond to restriction. Cuspidal automorphic representation or Eisensteries representations are infinite-dimensional, but $M$ automorphic representations are one-dimensional. On the other hand, there are positive examples where useful things can be said, e.g. Labesse-Langlands for $SL(2) \subset GL(2)$ .
Also this seems relevant: Restriction map between spaces of automorphic forms
