New Proof of the Fundamental Lemma

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.

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