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)?