Does local Langlands functoriality preserve genericity?

Question

Let $G=Sp_4$ over a p-adic field $F$. After Gan-Takeda's work, the local Langlands correspondence for $G$ is known. Thus we have the local functoriality from $G$ to $GL_5$. Do we know that this local Langlands correspondence preserve the genericity?

More precisely, let $\pi$ be an irreducible generic representation of $Sp_4(F)$, and $\phi: W_F'\rightarrow SO_5(\textbf{C})$ be the corresponding Langlands parameter determined by Gan-Takeda. Consider the Langlands parameter $\phi': W_F'\rightarrow SO_5(\textbf{C})\rightarrow GL_5(\textbf{C})$ by composing $\phi$ with the embedding of $SO_5$ into $GL_5$. Let $\pi'$ be the representation of $GL_5(F)$ corresponding to $\phi'$ by the local Langlands correspondence for $GL$. Do we know $\pi'$ is generic?

This question makes sense for general reductive group $G$ once we know the local Langlands correspondence for $G$. What is the general picture (or conjecture) for a general group $G$? Thanks in advance.

Answer 1

The general conjectural picture is the Gross-Prasad conjecture, found in Section 2 of Gross and Prasad's paper "On the decomposition of a representation of $SO_n$ when restricted to $SO_{n−1}$."

The notion of "generic representation" makes sense for a quasisplit group $G$, though one should fix a nondegenerate character of $U$ to be more precise. Anyways, let $\phi : W' \rightarrow {}^L G$ be a Weil-Deligne parameter, with ${}^L G$ the L-group of $G$. Let $Ad : G \rightarrow GL({\mathfrak g})$ be the adjoint representation with ${\mathfrak g}$ the Lie algebra of ${}^L G$. Then Gross and Prasad (and Rallis has a role in this too) state the following

Conjecture: The (conjectural) L-packet $\Pi_\phi$ contains a generic representation if and only if $L(Ad \circ \phi, s)$ is regular at $s=1$.

Note: It seems to me that if $G$ is not semisimple, one should look at the adjoint representation on the semisimple part of $\mathfrak g$, but maybe I'm missing something.

This seems like the right way to think about genericness on the Galois side of the Langlands correspondence, and there has been much progress on the Gross-Prasad conjecture for classical groups. See for example the survey of Wee Teck Gan. The Gross-Prasad conjecture is known for orthogonal groups, for example.

The question of whether a generic L-packet lifts to a generic L-packet, in some instance of functoriality, may be answered by considering suitable adjoint L-functions if one assumes the Gross-Prasad conjecture.

Answer 2

This is an attempt to give a family of counterexamples, somewhat following Victor Protsak's comment.

The basic idea is: let $\phi:W_F'\rightarrow SO_4$ be a generic Langlands parameter. Then I think $\phi^{Sp_4}:W_F'\rightarrow SO_4\subset SO_5$ should be usually generic, while $\phi^{GL_5}:W_F'\rightarrow SO_4\subset SO_5\subset GL_5$ is always non-generic as it factors through $GL_4\subset GL_5$.

How do we really find such an $\phi$? I am not familiar with the LLC for $Sp_4$ at all, but there are examples that can work. Here is one: let $p\not=2$, $T$ be a two-dimensional unramified anisotropic torus, and $\chi:T\rightarrow\mathbb{C}^{\times}$ a tamely ramified character that is "in general position." By LLC for tori we get $\phi_{\chi}:W_F'\rightarrow {}^LT$. One can embed ${}^LT\subset W_F\times SO_4\subset W_F\times SO_5$, so that $\phi_{\chi}^{Sp_4}$ is generic; it will be a Langlands parameter corresponding to the depth-zero supercuspidals of DeBacker and Reeder.