Let $G$ be a (split) reductive group over a $p$-adic field $F$, and $\mathbf{1}$ the trivial representation of $G(F)$. Under the (conjectural) Langlands correspondence, this should correspond to an $L$-parameter
$$
\varphi:WD\_F\longrightarrow G^\vee(\mathbb{C}),
$$
where $WD\_F$ is the Weil-Deligne group and $G^\vee(\mathbb{C})$ is the dual group of $G$. But what is this $\varphi$? Is it the trivial map? It doesn't seem to be so. If not, then is there any concrete way to write it down? Also if $\varphi$ is not the trivial map, then which irreducible admissible representation of $G(F)$ corresponds to the trivial map.


Also I have the same question for the corresponding $A$-parameter
$$
\psi:WD\_F\times\operatorname{SL}\_2(\mathbb{C})\longrightarrow G^\vee(\mathbb{C}).
$$



How to assign the $L$- and $A$-parameters for the trivial representation
The L-parameter of the trivial representation is the norm map $W\_F \to \mathbf{C}^\times$ composed with the cocharacter $\mathbf{C}^\times \to G^{\vee}(\mathbf{C})$ given by the half-sum of positive coroots. So e.g. for $GL\_n$ it sends $w \in W\_F$ to $\mathrm{diag}(|w|^{(n-1)/2},|w|^{(n-3)/2},\dots,|w|^{(1-n)/2})$.


The trivial L-parameter corresponds to the (irreducible!) normalized parabolic induction $i\_{B}^{G}(1)$, where $B$ is a(ny) Borel.


Your second question isn't well-posed - a representation can live in multiple A-packets. However, the trivial representation should certainly live in the A-packet attached to the Arthur parameter which is trivial on $WD\_F$ and embeds the Arthur $SL\_2$ as a principal $SL\_2$ inside $G^{\vee}$.


I hope this helps!


