On the Symmetric Square. Unstable Twisted Characters

We provide a purely local computa t ion of the (elliptic) twisted (by "transpose-inverse") character of the representat ion w = I (1) of PGL(3) over a p-adic field induced from the trivial representat ion of the maximal parabolic subgroup. This computa t ion is independent of the theory of the symmetr ic square lifting of [IV] of automorphic and admissible representat ions of SL(2) to PGL(3). It leads see [FK] to a proof of the (unstable) fundamental l emma in the theory of the symmetr ic square lifting, namely tha t 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]. This work continues the paper [FK], whose notations we use. Our aim is to prove Proposition 1 of [FK] without using Theorem 0 there. Namely we provide a purely local computation of the twisted character of ~r = I(1). Our model of ~r is that of [FK], where the twisted character X, is computed directly and locally but only for the anisotropic twisted conjugacy class 5' (see [FK], proof of Received January 18, 2002 307 308 Y.z. FLICKER AND D. ZINOVIEV Isr. J. Math. Proposition 1). In [FK] the value on the isotropic twisted conjugacy class 5 is deduced from the global Proposition 2.4 of [IV] recorded in [FK] as Theorem 0 which asserts that X.(5) = -X~ ((f'). While the proof of Proposition 2.4 of [IV] is independent of the results of [FK] (Theorems 1, 2, 3, 3 r, which follow from Proposition 1), it is global, and so might lead some readers to worry that a vicious circle is created. Moreover, the proof of this global result requires heavy machinery. Here we provide a purely local proof of Proposition 1 of [FK], and consequently make the results of [FK] independent of Proposition 2.4 of [IV] (= Theorem 0 of [FK]). Of course the conventional approach is to deduce the character computation of [FK], Proposition 1, on using the global trace formula comparison ([IV]) which is based on the fundamental lemma, proven purely locally in [V]. The novel approach of [FK] which we complete here is in reversing this perspective, and using the global trace formula to prove the (unstable) fundamental lemma from a purely local computation of the twisted character in a special case. Further, an independent, direct computation of the very precise character calculation gives another assurance of the validity of the trace formula approach to the lifting project. It will be interesting to develop this approach in other lifting situations, especially since our technique is different from the well-known, standard techniques of trace formulae and dual reductive pairs. A first step in this direction was taken in our work [FZ], where the twisted by the transposeinverse involution character of a representation of PGL(4) analogous to the one considered here, is computed. The situation of [FZ] is new, dealing with the exterior product of two representations of GL(2) and the structure of representations of the rank two symplectic group. Such character computations are not yet available by any other technique. However, the computations of [FZ] although elementary are involved, as they depend on the classification of IF] of the twisted conjugacy classes in GL(4). This is another reason for the present work, which considers the initial non trivial case of our technique where the computations are still simple and can clarify the method. We believe that our methods, pursued in [FZ] in a more complicated case, would apply in quite general lifting situations, in conjunction with, and as an alternative to the trace formula. Proposition 1 of [FK] asserts that if i is the trivial PGL(2,F)-module, lr = I(1) is the PGL(3,F)-module normalizedly induced from the trivial representation of the maximal parabolic subgroup (whose Levi component is GL(2,F)) , and 5 is a a-regular element of PGL(3,F) with elliptic regular norm ~l = N15, then Vol. 134, 2003 ON THE SYMMETRIC SQUARE 309