## Current Developments in Mathematics

- Current Developments in Mathematics
- Volume 2009 (2010), 1-20.

### Survey on the Fundamental Lemma

#### Abstract

This is a survey on the recent proof of the fundamental lemma. The
fundamental lemma and the related transfer conjecture were formulated by
R. Langlands in the context of endoscopy theory for automorphic representations
in "L-indistinguishability for SL(2)," *Canad. J. Math.* 31
(1979), no. 4, 726–785. Important arithmetic applications follow from the endoscopy
theory, including the transfer of automorphic representations from classical
groups to linear groups and the construction of Galois representations
attached to automorphic forms via Shimura varieties. Independent of applications,
endoscopy theory is instrumental in building a stable trace formula
that seems necessary to any decisive progress toward Langlands’ conjecture
on functoriality of automorphic representations.

There are already several expository texts on endoscopy theory and in
particular on the fundamental lemma. The original text and articles
of Kottwitz are always the best places to learn the theory. The
two introductory articles to endoscopy, one by Labesse, the other
written by Harris for the Book project are highly recommended. So are the
reports on the proof of the fundamental lemma in the unitary case written
by Dat for Bourbaki and in general written by Dat and Ngô Dac for
the Book project. I have also written three expository notes on Hitchin
fibration and the fundamental lemma : "Fibration de Hitchin et structure endoscopique de la formule des traces,"
*International Congress of Mathematicians*. Vol. II, 1213–1225, Eur. Math. Soc., Zrich,
2006, reports on endoscopic structure
of the cohomology of the Hitchin fibration, "Vietnamese congress of mathematicians" (2008) is a more gentle introduction
to the fundamental lemma, and "Decomposition theorem and abelian fibration," reports on the support theorem, a key
point in the proof of the fundamental lemma written for the Book project.
The survey follows the same plan as "Vietnamese congress of mathematicians" (2008) but more details have been added.

This report is written when its author enjoyed the hospitality of the Institute for Advanced Study in Princeton. He acknowledges the support of the Simonyi foundation and the Monell Foundation during his stay in the Institute.

#### Article information

**Source**

Current Developments in Mathematics, Volume 2009 (2010), 1-20.

**Dates**

First available in Project Euclid: 23 April 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.cdm/1335196621

**Mathematical Reviews number (MathSciNet)**

MR2757357

**Zentralblatt MATH identifier**

1238.11059

#### Citation

Châu, Ngô Bao. Survey on the Fundamental Lemma. Current Developments in Mathematics 2009 (2010), 1--20. https://projecteuclid.org/euclid.cdm/1335196621