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UNSTABLE TWISTED CHARACTERS

We provide a purely local computation of the (elliptic) twisted (by "transpose-inverse") character of the representation � = I(1) of PGL(3) over a p-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of (IV) of automorphic and admissible representations of SL(2) to PGL(3). It leads - see (FK) - to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in (FK) is the unstable one. A direct local proof of the fundamental lemma is given in (V).