
Vietnam
Journal of Mathematics 37:2&3 (2009) 127140

Report on the Proof of some Conjectures on Orbital
Integrals in Langlands' Program

Ngo Bao Chau

Abstract. Robert Langlands has formulated a
series of conjectures in local harmonic analysis known as the fundamental
lemma and the transfer conjectures. Though the statements are complicated,
these statements are entirely elementary and of combinatorial nature. They
become more notorious because of their difficulty and also of some deep
theorems in representation theory, number theory and arithmetic algebraic
geometry that rely thereon. The proof we have now of their conjecture, due
to the effort of many mathematicians, is based on local harmonic analysis,
ArthurSelberg's trace formula but surprisingly enough also on rather
involved algebraic geometry of certain moduli space which has origin from
mathematical physics.

In this report, I will recall the basics about orbital
integrals, the natural places in mathematics where we encounter with them,
the fundamental lemma and of the transfer conjecture which is stated in a
precise form only in certain cases. After surveying different contributions
to the solution of these conjectures, I will focus on certain algebraic
varieties that play a central role in the understanding of nonarchimedean
orbital integrals.


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by Vietnam Academy of Science and Technology & Vietnam Mathematical
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Published
by Springer since January 2013

