ON THE SYMMETRICAL SQUARE - UNIT ELEMENTS

A twisted analogue of Kazhdan's decomposition of compact elements into a commuting product of topologically unipotent and absolutely semi-si mple elements, is developed and used to give a direct and elementary p roof of the Langlands' fundamental lemma for the symmetric square lift ing from SL(2) to PGL(3) and the unit element of the Hecke algebra. Th us we give a simple proof that the stable twisted orbital integral of the unit element of the Hecke algebra of PGL(3) is suitably related to the stable orbital integral of the unit element of the Hecke algebra of SL(2), while the unstable twisted orbital integral of the unit elem ent on PGL(3) is matched with the orbital integral of the unit element on PGL(2). An Appendix examines the implications of Waldspurger's fun damental lemma in the case of endolifting to the theory of endolifting and that of the metaplectic correspondence for GL(n).