Why is Langlands functoriality usually related with period integral in a third group?

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.

• 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. Commented Nov 12, 2019 at 21:53
• 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. Commented Nov 12, 2019 at 22:30
• @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. Commented Nov 13, 2019 at 0:12

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, ...

• 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. Commented Nov 13, 2019 at 19:18

I'm not close to familiar enough with the references you cite or examples you ask about to address them, but here's a picture coming out of Sakellaridis and Venkatesh [SV]. Let us label a period not by a subgroup $$H\subset G$$ but by a spherical $$G$$-variety $$X$$, which could be homogeneous $$X=G/H$$ as in the Gan-Gross-Prasad case, the Hecke case, the group case discussed in Will Sawin's comment and many other interesting ones, or not. In fact it's better to replace $$X$$ by the hamiltonian $$G$$-space $$M$$ which could be $$T^*X$$, this puts on an equal footing other periods like Whittaker and the theta correspondence, but let's not do that here.

Then the theory developed by Knop, Gaitsgory-Nadler, Sakellaridis-Venkatesh, Knop-Schalke etc attaches to $$X$$ a subgroup $$G_X^\vee\subset G^\vee$$ [or in general a map to $$G^\vee$$ refining said subgroup]. One moral of the book [SV] is that this subgroup, or better yet the associated homogeneous space, controls much of the local harmonic analysis on $$X$$ as well as the global properties of $$X$$-periods. (In fact to get "complete" control of both questions one should also take into account an Arthur SL2 constructed by [SV] and a distinguished representation of $$G_X^\vee$$, which is the topic eg of the recent paper of Sakellaridis-Wong). e.g. the $$X$$-distinction of representations of $$G$$ (locally) or automorphic forms (globally) is controlled by factorization of their Langlands (or Arthur) parameters through $$G_X^\vee \times SL_2\to G^\vee$$.

In any case let me call $$G_X$$ the Langlands dual of $$G_X^\vee$$, which is in general nothing like a subgroup of $$G$$. (Hopefully this is the $$G'$$ of the paper you cite, but I'm very ignorant of that whole topic so can't evaluate.) Thus we have a Langlands (or Arthur) functoriality problem, from $$G_X$$ to $$G$$, which is related by [SV] to harmonic analysis on $$X$$ and $$X$$-periods.

In fact you can think of some of Knop's spectacular work on spherical varieties (or more general group actions) as expressing aspects of this functoriality -e.g., his Harish-Chandra homomorphism can be paraphrased as saying that the center of the enveloping algebra of $$G_X$$ appears naturally as invariant differential operators on $$X$$ -- in fact harmonic analysis on $$X$$ looks like it's "induced" from a $$G_X$$-space, even though $$G_X$$ by no means acts on $$X$$.