Twisted character of a small representation of ${\rm PGL}(4)$

We compute by a purely local method the elliptic $\theta$-twisted character $\chi_\pi$ of the representation $\pi=I_{(3,1)}(1_3)$ of ${\rm PGL}(4,F)$. Here $F$ is a $p$-adic field; $\theta$ is the “transpose-inverse” automorphism of $G={\rm PGL}(4,F)$; $\pi$ is the representation of ${\rm PGL}(4,F)$ normalizedly induced from the trivial representation of the maximal parabolic subgroup of type $(3,1)$. Put $C=\{(g_1,g_2)\in{\rm GL}(2)\times{\rm GL}(2)\colon\det(g_1)=\det(g_2)\}/\mathbb G_m$ ($G_m$ embeds diagonally). It is a $\theta$-twisted elliptic endoscopic group of ${\rm PGL}(4)$. We deduce from the computation that $\chi_\pi$ is an unstable function: its value at one twisted regular elliptic conjugacy class with norm in $C=C(F)$ is minus its value at the other class within the twisted stable conjugacy class, and 0 at the classes without norm in $C$. Moreover $\pi$ is the unstable endoscopic lift of the trivial representation of $C$.
Naturally, this computation plays a role in the theory of lifting from $C(=``SO(4)'')$ and ${\rm PG}_p(2)$ to $G={\rm PGL}(4)$ using the trace formula, to be discussed elsewhere ([F']).
Our work develops a 4-dimensional analogue of the model of the small representation of ${\rm PGL}(3,F)$ introduced with Kazhdan in [FK] in a 3-dimensional case. It uses the classification of twisted stable and unstable regular conjugacy classes in ${\rm PGL}(4,F)$ of [F], motivated by Weissauer [W]. It extends the local method of computation introduced by us in [FZ]. An extension of our work here to apply to similar representations of ${\rm PGL}(4,F)$ whose central character is not trivial has recently been given in [FZ'].