A (partial) explanation of the fundamental lemma and Ngo’s proof September 24, 2009
Posted by Joel Kamnitzer in Algebraic Geometry, geometric Langlands, Number theory, representation theory, things I don't understand.
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I would like to take Ben up on his challenge (especially since he seems to have solved the problem that I’ve been working on for the past four years) and try to explain something about the Fundamental Lemma and Ngo’s proof. In doing so, I am aided by a two expository talks I’ve been to on the subject — by Laumon last year and by Arthur this week.
Before I begin, I should say that I am not an expert in this subject, so please don’t take what I write here too seriously and feel free to correct me in the comments. Fortunately for me, even though the Fundamental Lemma is a statement about p-adic harmonic analysis, its proof involves objects that are much more familiar to me (and to Ben). As we shall see, it involves understanding the summands occurring in a particular application of the decomposition theorem in perverse sheaves and then applying trace of Frobenius (stay tuned until the end for that!).
First of all I should begin with the notion of “endoscopy”. Let 
be two reductive groups and let 
be there Langlands duals. Then 
is called an endoscopic group for 
if 
is the fixed point subgroup of an automorphism of 
. A good example of this is to take 
, 
. At first glance these groups having nothing to do with each other, but you can see they are endoscopic since their dual groups are 
and 
and we have 
.
As part of a more general conjecture called Langlands functoriality, we would like to relate the automorphic representations of to the automorphic representations of all possible endoscopic groups . Ngo’s proof of the Fundamental Lemma completes the proof of this relationship.
Let me be more precise now. Assume now that is defined over 
. The automorphic representations of are those irreducible representations of 
that occur in 
(here 
is the adeles). We can study this representation using the trace formula which relates the character of this representation to certain “orbital integrals”. More precisely we have we have an expression like:
where 
is a smooth compactly supported function on and 
is its character for the representation .
In the LHS, 
is something like an conjugacy class in 
, and 
is an orbital integral, ie an integral over the conjugacy class given by 
.
In order to related automorphic representations of and , we will relate the corresponding left hand sides of the trace formula. Hence one obtains a conjectural relationship between the orbital integrals for and the orbital integrals for for all endoscopic 
. This is the Fundamental Lemma.
Let us now consider the function field setting. This means that we take to be defined over a finite field 
and replace 
by a function field 
of a curve 
defined over 
. Then we can speak about the Fundamental Lemma in this context. Surprisingly by work of Waldspurger this implies the Fundamental Lemma in the number field setting (how can this be?). Ngo has proved the Fundamental Lemma in this function field setting. The advantage of this setting is that it is easier to use geometric means.
Let me now switch gears a bit and talk about the Hitchin fibration. Fix a line bundle 
on 
. Let 
deonte the moduli space of pairs 
, where 
is a principal 
bundle on and 
is a section of 
. For 
this would be a rank n vector bundle along with a morphism 
.
The moduli space 
comes with a map 
to an affine space 
where 
is the rank of and 
are the exponents of the group. This map is defined using the basic invariant polynomials of (which have degrees 
).
How is this relevant to the Fundamental Lemma? The key observation of Ngo (or maybe Ngo and Laumon) is that the number of points in a fibre of (a Hitchin fibre) is a certain sum of orbital integrals. The reason is that Hitchin fibres are related to affine Springer fibres and the relation between affine Springer fibres and orbital integrals is straightforward from the definitions. (Goresky-Kottwitz-MacPherson have extensively studied the Fundamental Lemma from the affine Springer fibre perspective.)
So, now we (or rather Ngo and Laumon) have reduced the fundamental lemma to the problem of relating the number of points of Hitchin fibres of to the number of points of Hitchin fibres for the endoscopic groups .
By Grothendieck, we know that the way to count the number of points of a variety is to take the trace of Frobenius acting on the cohomology of the variety over the algebraic closure. Hence, we can simply try to relate the cohomology of the Hitchin fibres for to those for .
Better yet, we can organize all these cohomologies together and study the pushforward of the constant sheaf under . There is a locus in 
, called 
over which is proper, so we have 
. When we have a proper map, the decomposition theorem applies and tells us that the pushforward of the constant sheaf 
is a direct sum of shifted perverse sheaves.
For any endoscopic group , there is an inclusion 
of 
into 
(because of the relationship of the dual groups, we have an inclusion of Weyl groups 
and only depends on the Weyl group). Ngo’s main theorem is that (roughly)
where ranges over the endoscopic groups of .
Taking stalks and then trace of Frobenius gives an equality of orbital integrals and hence the Fundamental Lemma.
One question for anyone who has made it so far: is there a relationship between this appearance of the Hitchin moduli space and its appearance in geometric Langlands (in the works of Beilinson-Drinfeld and Kapustin-Witten)?
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Great summary Joel!
“Surprisingly by work of Waldspurger this implies the Fundamental Lemma in the number field setting (how can this be?).”
An argument I heard about is via logic. Roughly speaking you can formulate both the p-adic and function field version of the fundamental lemma in some axiomatic system (first-order language), and if you can prove one that implies the other. This is called the transfer principle, check it out here http://www.dma.ens.fr/~loeser/transfer.pdf
“The key observation of Ngo (or maybe Ngo and Laumon) is that the number of \mathbb{F}_q points in a fibre of f (a Hitchin fibre) is a certain sum of orbital integrals.”
I think it is a product of orbital integrals. Namely the geometrical observation is that the Hitchin fiber modulo its Picard group as a stack is the product of various affine Springer fibers modulo their Picard. The number of points on the affine Springer fibers modulo its Picard then is identified with those orbital integrals.
“Ngo’s main theorem is that (roughly)
Ngo’s main theorem is more precise as it tells you how exactly the left hand side decomposes. Namely Ngo constructs a finite group scheme
over 
, namely the group of connected components of the Picard group scheme, which acts on the Hitchin fibration with a dense open orbit. This action will filter through an action of 
on the cohomology
and the decomposition into the various characters of 
will give you the right hand side (again with precisely identified components.)
Also I believe that $\latex \Gamma$ is trivial for
(over 
) thus in that case Ngo’s theorem and so the fundamental lemma ought to be vacuous.
Thanks for all the info, Tamas.
Maybe I should have also mentioned is that there are some nice recent papers by Zhiwei Yun on this subject (namely on decomposition of the cohomology of the Hitchin space).
The video of the ICM 2010 lecture of Ngo Bao Chau is now online.
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