A new proof of the Fundamental Lemma has been announced by

Groechenig, Wyss, and Ziegler [GWZ].

The fundamental lemma is an identity between integrals on p-adic

reductive groups that was conjectured by Langlands in his Paris

lectures on the stable trace formula (1979). The name

“*fundamental lemma*” was a misnomer, because for decades it was only a

conjecture. These conjectures were later put into more precise form

by Langlands and Shelstad in 1987, then extended by Kottwitz and Shelstad (Foundations of Twisted Endoscopy, 1999) and Arthur in the twisted case.

The fundamental lemma was studied intensively by numerous researchers

for decades before a solution was found. Over the years the

fundamental lemma was transformed successively as a question about

counting points in Bruhat-Tits buildings (Langlands, Rogawski),

asymptotics and Igusa theory (Langlands and Hales), affine Springer

fibers (Kazhdan and Lusztig) and equivariant cohomology (Goresky,

Kottwitz, MacPherson), compactified Jacobians (Laumon, Ngo), and the

Hitchin fibration, stacks, and perverse sheaves (Ngo). Over time,

Waldspurger provided major simplifications of the original

conjecture. This list of mathematicians includes many that I greatly

admire.

These identities were finally established by Ngo Bao Chau (and by

Chaudouard-Laumon in the twisted case) in a spectacular proof that led

to his Fields medal.

Although I am no longer an active researcher in the Langlands program,

Robert Langlands was my thesis advisor, and my primary research

interest was the fundamental lemma and related identities during the

years 1984-1993. I gave a Bourbaki report on Ngo’s proof of the

fundamental lemma.

The fundamental lemma provides the identities between integrals that

are absolutely crucial for the existence of the stable form of the

Arthur-Selberg trace formula. The stable trace formula has become a

major tool in the study of automorphic representations, which lie at

the heart of the Langlands program.

Many applications in number theory follow from the fundamental lemma,

through the stable trace formula. For example, one of the earliest

cases of the fundamental lemma — base change for GL(2) — yielded

a trace formula that led to the Langlands-Tunnell theorem, an

essential ingredient in the proof of Fermat’s Last Theorem. Arthur’s

book on the classification of automorphic representations of classical

groups is another major application of the trace formula and

fundamental lemma.

As mentioned above, a recent preprint GWZ on the arXiv gives a new proof, which we now describe.

## Hitchin fibration

One of the main ideas that led to a solution to the fundamental lemma

was Ngo’s realization that the Hitchin fibration is the natural global

analogue of affine Springer fibers. The starting point for the GWZ

proof of the fundamental lemma is also the Hitchin fibration. In

this, they do not differ from Ngo. They also follow Ngo in making

heavy use of a Picard stack that acts on the Hitchin fibration.

To appreciate the new proof, it is necessary to look at the details of

Ngo’s proof to see how it has been simplified by GWZ. Notably,

the perverse sheaves, support theorems, and decomposition theorem, which were essential to Ngo’s approach, have been entirely eliminated. (The support theorems

take up a very technical 25 page section in Ngo.) In fact, in the GWZ

proof there are no sheaves at all, and only light use of cohomology

(for the Brauer group and Tate duality, which are needed even to state

the fundamental lemma).

In my mind, the ugliest part of Ngo’s (otherwise beautiful) proof came in his use of a

stratification by a numerical invariant. The construction of the

Hitchin fibration depends on a line bundle, and Ngo’s proof relies on

complicated relations between the stratification and the line bundle

as the degree of the line bundle tends to infinity. This very

technical part of the proof of the fundamental lemma has been

eliminated in the GWZ proof.

Another part of Ngo’s proof that has been eliminated by GWZ is the

explicit calculation of orbital integrals. Overall, the GWZ proof is

simpler in two ways: it eliminates the most complex machinery (such as

the decomposition theorem) and the most complex calculations (such

stratification estimates). However, the Hitchin fibration and stacks

are still needed in both proofs.

## GWZ proof of the fundamental lemma

The GWZ approach originated with work on the topological mirror

symmetry conjecture for the smooth moduli space of Higgs bundles,

which uses p-adic integration to prove identities of stringy Hodge numbers. They realized that similar techniques should work to give a new proof of the fundamental lemma.

Ngo’s work shows the number of points (on Hitchin fibers over finite fields)

are related on different groups. The key idea of the GWZ proof is

that these point counting problems can be expressed as a p-adic

integral (over the Hitchin fibration). The advantage

of working with integrals is that they are insensitive to sets of

measure zero, and this allows badly behaved Hitchin fibers to be

ignored. This is crucial, because the badly behaved fibers are the

source of all the woes. Some of the ideas here have been influenced

by motivic integration, especially a paper by Denef

and Loeser on Batyrev’s form of the McKay correspondence.

In formulating the fundamental lemma as a p-adic identity, the GWZ

proof is the latest zag in a curious series of zig-zags between global

fields and local fields: the trace formula

(global) led to the fundamental lemma and affine Springer fibers

(local), which was reformulated on the Hitchin fibration (global), as

identities of p-adic integrals (local).

The fundamental identity in GWZ takes the form of a remarkable duality

of integrals between a reductive group G and its Langlands dual G’:

`(1) ∫_M(G,t,O) χ(s) dG = ∫_M(G',s,O) χ(t) dG'`

(after dropping some subscripts and superscripts) where s and t

run over appropriate character lattices, O is the set of

integer points of a local field, M(G,t) are twists of the Hitchin

fibration M on G (and G’), and χ are functions defined through the Brauer

group (related to the κ in Langlands’s original formulation). This identity directly implies the fundamental lemma, and it

is cleaner than other ways of stating the fundamental lemma. For that

reason, I would propose that we accept the GWZ formula as the

*primal* form of the fundamental lemma.

The fundamental identity (1) when the functions χ are constant

is expressing that dual abelian varieties have the same volume. When χ is nontrivial, the identity is saying that a nontrivial character integrated over an abelian group is zero. Thus, the GWZ paper exposes the bare structure of

the fundamental lemma as a duality for abelian varieties.

In summary, the GWZ paper gives a new proof of the fundamental lemma,

which bypasses the most difficult parts of the earlier proof and exposes

its essential structure.