A twisted analogue of
Kazhdan’s decomposition of compact elements into a commuting product of
topologically unipotent and absolutely semi-simple elements, is developed and used
to give a direct and elementary proof of the Langlands’ fundamental lemma for the
symmetric square lifting from SL(2) to PGL(3) and the unit element of the Hecke
algebra. Thus 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 element on PGL(3) is matched with the
orbital integral of the unit element on PGL(2). An Appendix examines the
implications of Waldspurger’s fundamental lemma in the case of endolifting to
the theory of endolifting and that of the metaplectic correspondence for
GL(n).
