2
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ It seems also reasonable to care first about the local situation before asking about the global representation theory. $\endgroup$
    – Marc Palm
    Nov 26 '13 at 10:14
3
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.