On the symmetric square: unstable local transfer

We prove the "fundamental lemma" for spherical functions with respect to the natural (induction) lifting from PGL(2) to PGL(3) which appears as the unstable counterpart of the stable symmetric-square lifting from SL(2) to PGL(3) (see [IV] for an introduction to this project, and [VI] for the final results). Thus spherical functions on PGL(2) and PGL(3) which correspond to each other by satisfying an elementary representation theoretic relation are shown to have matching orbital integrals. The proof of this local statement is based on an application of the global trace formula.