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In the introduction of "PERIODS OF AUTOMORPHIC FORMS "by HERVE JACQUET, EREZ LAPID, and JONATHAN ROGAWSKI, they said

"In many cases, it should be possible to characterize the $H$-distinguished cuspidal representations as images with respect to a functorial transfer to $G$ from a third group $G'$"

here $G$ is a reductive group over a number field $F$ , and $H \subseteq G$ is a subgroup obtained as the fixed point set of an involution $\theta$.

The case $(GL_n(E),GL_n(F))$ and $(GL_n(E),U_n(F))$ is well-studied by Jacquet, here $E/F$ is a quadratic extension. For general Galois model, there is a conjecture by Prasad (see https://arxiv.org/abs/1512.04347).

There are examples beyond the involution case, e.g the nonvanishing of Shalika period integrals is characterized in terms of functorial transfers from $GSpin(2n + 1)$.

There are also local analogues for the story. The question is (at least over $p-$adic field):

Why is Langlands functoriality usually related with period integral in a third group? What do we know beyond Galois case? What's the relation with relative Langlands program by Sakellaridis and Venkatesh?

Toy model: an irreducible cuspidal representation of $GL_n(\mathbb F_{q^2})$ is distinguished by $U_n(\mathbb F_q)$ iff it comes from $GL_n(\mathbb F_q)$ by Shintani lifting.

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    $\begingroup$ Given $H$ a subgroup of $G$, there may be no natural way to write the $L$-group of $H$ as a subgroup of the $L$-group of $G$, so there would be no way to relate the period integral to functoriality from $H$ as that does not exist. $\endgroup$ – Will Sawin Nov 12 '19 at 21:53
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    $\begingroup$ One easy test case is when $G = H \times H$. Then a representation $\pi_1 \otimes \pi_2$ has an $H$-invariant vector (maybe in some completion) if and only if $\pi_2 = \pi_1^\vee$, which corresponds to functoriality from a third group, isomorphic to $H$. This is the untwisted variant of your toy model. I think you can see a lot of the issues (like the analytic subtleties in the definition of the period integral) already in this case. $\endgroup$ – Will Sawin Nov 12 '19 at 22:30
  • $\begingroup$ @WillSawin Thank you. Of course, the question is not for a subgroup of the group itself (but for the Galois model one can do this). In general, functoriality starts with a map ${}^LH \rightarrow {}^LG$, and one hopes to characterize the image by using some period integral on another group (e.g the example of Shalika period integral as above). There are some convergence issues, in the local setting if we assume the representation is supercuspidal then this is OK. $\endgroup$ – sawdada Nov 13 '19 at 0:12
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A lot is known--too much to try to summarize--and I think this philosophy came about after seeing numerous examples, beginning with Harder-Langlands-Rapoport (base change for GL(2)), and thinking about the connection with L-functions.

Namely, in many instances one knows that images of functorial lifts can be detected by poles or nonvanishing of certain $L$-functions at special values. For instance, often images of functorial lifts can be detected via factorizations of $L$-functions for functorial lifts---e.g., to relate to Will Sawin's comment for a lift from GL(n) to GL(n) $\times$ GL(n), the zeta function divides $L(s, \pi \times \check \pi)$ so the latter has a pole at $s=1$. In addition, theta lifts are often nonvanishing only when a certain $L$-value is non-zero.

Now one often has integral representations for $L$-functions, and one can often detect poles/nonvanishing of $L$-functions at special values via period integrals. In addition, one can often compare periods on different groups, as in much of Jacquet's work and the Gan-Gross-Prasad conjectures. Sakellaridis-Venkatesh is an attempt to build a general theory for how periods are related on different groups. They primarily study the local question, but do make some global conjectures at the end as well.

Some places you can look for instances of what is known: surveys/papers on integral representations, poles of $L$-functions, theta correspondence and descent; Jacquet's papers; introduction to Sakellaridis-Venkatesh, ...

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  • $\begingroup$ Thank you! The local story is still mysterious (the relation between local Langlands correspondence, local L function and period integrals seems not obvious), I will have a look on those papers. $\endgroup$ – sawdada Nov 13 '19 at 19:18

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