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Abstract
Recent work of Altu\u{g} continues the preliminary analysis of Langlands' Beyond Endoscopy proposal for GL(2) by removing the contribution of the trivial representation to the trace formula using a Poisson summation formula. We show that Altu\u{g}'s method of smoothing real elliptic orbital integrals by an approximate functional equation extends to GL(n). We also discuss the case of an arbitrary reductive group, and remaining obstructions for applying Poisson summation.
We give a short introduction to Beyond Endoscopy, a proposal by Langlands for attacking the general principle of functoriality. We shall try to motivate the proposal by emphasizing its structural similarities with the actual theory of endoscopy. We shall then discuss a few of the many problems that will need to be solved, some of which are suggested by the recent work of A. Altuğ.
With analytic applications in mind, in particular Beyond Endoscopy ([13]), we
initiate the study of the elliptic part of the trace formula. Incorporating the
approximate functional equation to the elliptic part we control the analytic
behavior of the volumes of tori that appear in the elliptic part. Furthermore
by carefully choosing the truncation parameter in the approximate functional
equation we smooth-out the singularities of orbital integrals. Finally by an
application of Poisson summation we rewrite the elliptic part so that it is
ready to be used in analytic applications, and in particular in Beyond
Endoscopy. As a by product we also isolate the contributions of special
representations as pointed out in [13].
We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a study of the decomposition of the l-adic cohomology of the Hitchin fibration into direct sum of simple perverse sheaves.
Jan 2010
199-243
E Frenkel
R P Langlands
B C Ngô
E. Frenkel, R. P. Langlands and B. C. Ngô, La formule des traces et la functorialité. Le
debut dun Programme, Ann. Sci. Math. Québec 34 (2010), 199-243.
Jan 1981
649-664
R Kottwitz
R. Kottwitz, Unstable orbital integrals on SL(3), Duke Math. J. (3) 48 (1981), 649-664.