Computation of a twisted character of a small representation of GL(3, E)

Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let x → x̄ be the involution of E over F. The representation π of GL(3, E) normalizedly induced from the trivial representation of the maximal parabolic subgroup is invariant under the involution σ(g) = J ^tḡ ^-1J. We compute - by purely local means - the σ-twisted character χ _pi ^σ of π. We show that it is σ-unstable, namely its value at one σ-regular-elliptic conjugacy class within a stable such class is equal to negative its value at the other such conjugacy class within the stable class, or zero when the σ-regular-elliptic stable conjugacy class consists of a single such conjugacy class. Further, we relate this twisted character to the twisted endoscopic lifting from the trivial representation of the "unstable" twisted endoscopic group U(2, E/F) of GL(3, E). In particular π is σ-elliptic, that is, χ _pi ^σ is not identically zero on the σ-elliptic set.