COMPUTATION OF A TWISTED CHARACTER OF A SMALL REPRESENTATION OF GL(3, E)

Abstract

Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let [Formula: see text] 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 [Formula: see text]. We compute — by purely local means — the σ-twisted character [Formula: see text] 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, [Formula: see text] is not identically zero on the σ-elliptic set.