The fundamental lemma (FL) is a key ingredient in the stabilization of

the Arthur-Selberg trace formula, which is part of the heavy machinery

used in the Langlands program. Ngo Bao Chau received a Fields medal

for his proof of the FL. His proof is an amazing mathematical

achievement.

The FL has come to my attention again, because of a new proof

of the FL posted on the arXiv. An interesting feature of this

proof is the way that non-standard and standard identities become intertwined.

## Twisted Endosopy

I became an assistant professor at the University of Chicago in 1990,

just down the hall from Kottwitz. He freely shared with me the early

versions of his research with Shelstad on the foundations of twisted

endoscopy as the manuscripts became available. Through Kottwitz’s

mentoring, I was one of the first to think about the general theory of

twisted endoscopy.

## Descent

The paper by Langlands and Shelstad on descent for transfer factors

was the first paper I was ever asked to referee. The paper was

related to my thesis work under Langlands, and I went over

their paper in detail. Descent shows that identities of

orbital integrals can be deduced from identities of orbital integrals

on smaller reductive groups, by taking centralizers.

## Combining Twisted Endoscopy with Descent

Clozel invited me to spend three months at Orsay in the fall of 1992.

Computing many twisted centralizers, I found that modulo standard

endoscopy, descent always gave the same conjectural stable identities

relating root systems G2 with G2, F4 with F4, or Bn with Cn,

exchanging root lengths. For GL(4) and GL(5), the

relevant identities are between B2 and C2. Calculations yielded

an elliptic curve for B2 and another

elliptic curve for C2. An isogeny between the elliptic curves

gave the transfer of orbital integrals.

I completed these calculations in time for my lecture in Orsay on

November 17, 1992, where I described the conjectural dualities of

stable orbital integrals exchanging root lengths for the non-simply laced

root systems, as well as the evidence provided by the elliptic curve

isogenies for B2-C2. By my records, that was the first public description of

the non-standard identities.

I repeated these non-standard conjectures in other lectures, notably

in lecture series at IAS in 1993 and 1995. at For example, my notes

for the third lecture at IAS in February 1993 contain the statement,

“Hope: the theory of twisted endoscopy is no more than standard

endoscopy combined with the stable dualities: (G2 ↔ G2), (F4 ↔ F4),

(Bn ↔ Cn), long roots ↔ short roots.” I never pushed these as my

conjectures because I have always viewed them as a direct consequence of

(and the most reasonable route to the proof of) the deeper conjectures

of Langlands, Kottwitz, and Shelstad. Nonetheless, a precise idea was

in place by late 1992.