Lapid–Prasad conjecture on distinguished representations and invariant L-packets

Let $G$ be a connected reductive algebraic group defined over a local non-archimedean field $F$, let $\theta:G\rightarrow G$ be an involution defined over $F$, and set $H=G^{\theta}$. Let $\pi$ be a smooth irreducible representation of $G(F)$. An irreducible representation is $H$-distinguished if it admits a nonzero $H$-invariant linear form.

Lapid–Prasad conjecture. If $\pi$ is $H$-distinguished, then the $L$-packet of $\pi$ is invariant under the functor

$$\tau\longmapsto \widetilde{\tau}^{\theta},$$

\nwhere $\widetilde{\tau}$ is the contragredient representation and $\tau^{\theta}$ is the twist by $\theta$.

This gives a necessary condition for distinction in the representation theory of symmetric pairs. The supplied status evidence says that the case of real Galois symmetric pairs was proved, while the finite-field setting is described as another natural setting in which the conjecture can be stated; the general local non-archimedean formulation is therefore recorded here as solved according to the supplied resolution status.