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Vietnam Journal of Mathematics 37:2&3 (2009) 127-140 

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, Arthur-Selberg'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 non-archimedean orbital integrals.

 

 

 

 

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