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
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.
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
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.