How to assign the $L$- and $A$-parameters for the trivial representation

Question

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 $$ \psi:WD_F\times\operatorname{SL}_2(\mathbb{C})\longrightarrow G^\vee(\mathbb{C}). $$

Answer 1

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!

Answer 2

This is an instance of the Satake correspondence. Recall that a smooth irreducible representation $\pi$ of $G(F)$ is unramified if $\pi$ has a $G(\mathcal O_F)$-invariant vector. Then the Satake correspondence claims that there is a bijection between the set of unramified representations of $G(F)$ and the orbit space $W\backslash T^\vee(\mathbb C)$.

Explicitly, given a $t\in T^\vee(\mathbb C)$, we may view it as a homomorphism $\chi_t\colon W_F\to T^\vee(\mathbb C)$ factoring through $\mathbb Z$. Then the normalized parabolic induction $\mathrm{i}_B^G(\chi_t)$ is such that $(\mathrm{i}_B^G\chi_t)^{G(\mathcal O_F)}$ is $1$-dimensional, so it has a unique unramified subquotient. Moreover, this construction is $W$-invariant. Conversely, given an unramified representation $\pi$ with a $G(\mathcal O_F)$-invariant vector $v$, we obtain a homomorphism $t\colon T(F)/T(\mathcal O_F)\to \mathbb C^\times$ by taking, for any $a\in T(F)$, the integral $\frac1{\mathrm{vol}\ G(\mathcal O_F)}\int_{G(\mathcal O_F)}\pi(k)\pi(a)vdk$. This is again $G(\mathcal O_F)$-invariant, and since the $\pi^{G(\mathcal O_F)}$ can be showed to be $1$-dimensional, it is of the form $t(a)v$. Now $t\colon T(F)/T(\mathcal O_F)\to\mathbb C^\times$ can equivalently be viewed as an element of $T^\vee(\mathbb C)$.

Since normalized parabolic induction $\mathrm{i}_B^G$ is off from the naive induction $\mathrm{Ind}_B^G$ by the half-sum of positive roots, and clearly $\mathbf1$ is an unramified subquotient of $\mathrm{Ind}_B^G(\mathbf 1)$, the half-sym of positive roots must be the L-parameter of $\mathbf 1$.