Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}\_n$.) Now $G(E)$ can be viewed in two different way: the $F$ points of the restriction of scalar $G\_1:=\operatorname{Res}\_{E/F}G$, and the $E$-points of just $G\_2:=G$. Their Langlands $L$-groups are respectively
$$
{^LG\_1}=(G^\vee(\mathbb{C})\times G^\vee(\mathbb{C}))\rtimes W\_F\qquad\text{and}\qquad{^LG\_2}=G^\vee(\mathbb{C})\times W\_E,
$$
where $G^\vee$ is the dual group of $G$ and for the semidirect product the nontrivial element in $W\_F/ W\_E$ acts by switching the two factors.


Now, assuming the local Langlands correspondence, for each irreducible admissible representation of $G(E)$, one can associate local Langlands parameters
$$
\varphi\_1:\operatorname{WD}\_F\longrightarrow{^LG\_1}\qquad\text{and}\qquad 
\varphi\_2:\operatorname{WD}\_E\longrightarrow{^LG\_2},
$$
where $\operatorname{WD}\_F$ and $\operatorname{WD}\_E$ are the Weil-Deligne groups.


I believe there must be a way to pass from $\varphi\_1$ to $\varphi\_2$. But how?



Two different local Langlands parameters for quadratic extension
This came up in a paper of mine not so long ago, and my coauthors and I were surprised that it wasn't made explicit in the standard references, so we wrote it out ourselves:


*Dembélé, Lassina; Loeffler, David; Pacetti, Ariel*, [**Non-paritious Hilbert modular forms**](https://doi.org/10.1007/s00209-019-02229-5), Math. Z. 292, No. 1-2, 361-385 (2019). [ZBL1446.11084](https://zbmath.org/?q=an:1446.11084).


See Proposition 1.3. (Strictly speaking we are describing the analogue for global fields, not local fields, but the recipe is the same.)


