Is this a subcase of the fundamental lemma?

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Let $F$ be a local field and $G= GL(n,F)$.

Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$O_\gamma^G( \phi) = \int\limits_{G_\gamma \backslash G} \phi( g^{-1} \gamma g) d g.$$

We can assume wlog that $\gamma$ is elliptic. Can we lift the elliptic orbital integral on $G$ to an elliptic orbital integral on $G' =GL(n,F')$, where $F' = F[X] / det(X-\gamma)$?

More precisely, can we find an $T:C_c^{\infty}(G) \rightarrow C_c^\infty(G')$ such that $$O_\gamma^G(\phi) = O_\gamma^{G'} ( T\phi).$$

I think for explicit computations, it is much more convenient to work on $G'$, where $\gamma$ is diagonalizable nad $G'_\gamma$ can be chosen to be the diagonal matrices.

What is the Reference? What is the Buzzword for this?

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1 Answer

I will offer some words on this, but only because no-one else has; I was holding out hoping that one of the more automorphic people would chip in. It might be worth taking much of the below with a pinch of salt.

So I've been trying to penetrate the "fundamental lemma" literature myself, and let me begin by showing my hand and saying that my current impression, that may be wrong, is that "the fundamental lemma" is not a well-defined mathematical statement, it is a principle that applies in many situations, and I think that mathematics is open to the situation where there will be precise mathematical statements formulated in the future, when people are studying situation $X$, and that people will call these statements "the fundamental lemma for $X$". For example, I think that perhaps in 50 years' time when people are proving general base change (rather than cyclic base change) there may be a "general base change fundamental lemma".

So this of course immediately raises the question -- what did Ngo Bao Chau do? Well, my perception is that he has proved the version of the fundamental lemma that shows up in the theory of endoscopy.

As I've already tried to make clear, I am not 100% sure on all of this. But let me muddle on regardless. My impression is that what the game looks like is this. There is the general functoriality principle, which says that if I have two connected reductive groups $G$ and $H$ over a global field, and a map between the $L$-group ${}^LH\to{}^LG$, then there should be some sort of way of transferring automorphic representations on $H$ to automorphic representations on $G$. Already this is a "principle" rather than a precisely-formulated statement, because I think there are lots of issues with packets and multiplicities, that certainly I don't understand, when trying to make a precise statement, and I am not sure I've seen a precise statement in the literature that ties up all the loose ends that I want to see tied up, other than some very weak ones that just demand compatibility at the unramified places and don't care about multiplicities at all.

But it turns out, when attempting to e.g. study the zeta functions of some Shimura varieties, that Langlands needed some very special cases of the functoriality principle, where $H$ was an endoscopic subgroup for $G$, and here he postulated a strategy for proving functoriality using the trace formula. The idea is that you try and stabilise the trace formula for $G$, whatever that means, and this involves, amongst other things, figuring out some way of transferring functions locally and matching the orbital integrals that come up. The upshot is that you relate the trace formula for $G$ to the trace formula for all the endoscopic subgroups for $G$.

So my perception is that, in its initial state, the situation was this: $G$ was a connected reductive group, now over a local field, $H$ an endoscopic subgroup (and part of the data of this endoscopic situation is that you're given a map ${}^LH\to{}^LG$) and there is supposed to be a "transfer" map, that maps functions on $G$ to functions on $H$. Even the existence of the transfer map is not at all clear in general, I don't think. For example Labesse-Langlands have to do some calculations (only a few pages, but some work) to prove that one can transfer functions from $SL(2)$ to a subtorus (this is the simplest example of endoscopy, I think).

So my impression is that the general notion of moving from functions on one group, locally, to functions on another, is called "transfer of functions". My understanding is that the transfer map is not at all well-defined, that sometimes one can characterise the image (as being functions whose orbital integrals vanish on some certain subgroup), and that at the end of the day the precise relationship you want between the function and its transfer can be quite complicated. I think that in general you want the orbital integral of one function to be the orbital integral of the other multiplied by a "fudge factor" whose definition is the key point of a 100-page paper by Langlands and Shelstad. One can already see these fudge factors in the $SL(2)$ case with Langlands-Labesse.

My understanding of what the fundamental lemma is, is the following: in the situation where $G$ and $H$ are unramified, one extra condition you could put on the "transfer of functions" map is that the identity element for the unramified Hecke algebra for $G$ gets sent to the identity element for the unramified Hecke algebra for $H$. Hence in this situation, "the fundamental Lemma", I think, boils down to the assertion that a certain volume equals a fudge factor that it takes 100 pages to define, multiplied by another volume.

I'm slowly getting to the point :-) I think I can answer one of your questions at least -- the "buzzword" you're looking for is not "fundamental lemma" but "transfer of functions" or perhaps "local transfer of functions". I think.

However, my understanding is that the situation you are looking at is not an "endoscopic situation". In particular I don't think that reading the complete works of Ngo Bao Chau will give you an answer to your question. You have groups $G$ and $G'$ and they're both $GL(n)$ but over different fields, so if I were trying to prove a hard global theorem and I needed the type of transfer that you're looking for, I would probably not be trying to stabilise the trace formula (indeed I think the trace formula for $GL(n)$ is already stable and that "$GL(n)$ has no endoscopy" in some sense) -- I would probably be trying to prove base change.

Now here are, for me, some BIG problems I would fear when attempting to try and get an answer to your question from the literature that I know about.

The first is that, when trying to prove global base change for $GL(n)$ for a global cyclic extension $L/K$, I attempt to find a relation between the trace formulas for $GL(n)$ over $L$ and over $K$ and then I attempt to start matching up terms etc etc, and the problem is that one trace formula is a sum over conj classes in $GL(n,L)$ and the other is a sum over $GL(n,K)$. As far as I know, people don't know how to relate these two sets in a natural way. So what they do is they relate conjugacy classes in $GL(n,K)$ to twisted conjugacy classes in $GL(n,L)$. I think the local story looks like this: say $E/F$ is now local and has cyclic Galois group. Given $\gamma$ in $GL(n,E)$ they take its "norm" in the most naive way (multiply it by its Galois conjugates) and get an element of $GL(n,F)$ and in this nice cyclic situation they can inject twisted conj classes ($x\sim \sigma(g)xg^{-1}$ with $\sigma$ a generator of the Galois group) in $GL(n,E)$ with conj classes in $GL(n,F)$. Using this trick they want to relate the usual trace formula for $GL(n,K)$ with a twisted trace formula for $GL(n,L)$, and to do this the transfer of functions they require is a twisted transfer! In particular, they need a machine which, given a function $a$ on $GL(n,E)$ spits out a function $b$ on $GL(n,F)$ such that the orbital integrals of $b$ equal certain twisted orbital integrals for $a$, possibly again multiplied by some fudge factors, but I believe that in this base change situation the fudge factors (which are I think not covered by the Langlands-Shelstad monster because this is not an endoscopic situation) are all 1 anyway.

The upshot is that in the Arthur-Clozel book, proving cyclic base change for $GL(n)$, you will see a definition of transfer of functions, which looks formally a bit similar to the thing you write above, but there are certain crucial differences:

1) if $a$ transfers to $b$ then an orbital integral for $b$ will equal a twisted orbital integral for $a$, rather than an orbital integral.

2) The twisted orbital integral for $a$ will be attached to an element $\gamma$ and the orbital integral for $b$ that it equals will be attached not to $\gamma$ but to its naive norm (which is a well-defined conj class, I believe, in $GL(n,F)$)

3) $E/F$ will ALWAYS be cyclic (or perhaps more generallu they will allow $E=F^n$ so $E$ is not actually a field but it's still etale over $F$).

Now I look at the question you're asking, and there are of course two ways of approaching it: the first is to try and get your hands dirty and write down the map between functions yourself. But it sounds to me like you're hoping that you can take another approach -- to get what you want from the literature. And what I am really scared by is that although what you write looks to me superficially like transfer of functions, you are in a situation where $F'$ is not in general a cyclic Galois extension of $F$, so you can throw away Arthur-Clozel, and you are not I think in an endoscopic situation either, so you can also throw away Ngo Bao Chau, and unfortunately I personally do not know what is left. Of course this will largely be my own ignorance and probably there are people who have thought about "transferring" in its own right, independent of links to functoriality. But I am now not sure where to point you.

Aah, the joyful tones of my daughter, who has apparently just woken up. What timing she has! I've gotta go, but I've said all I can say anyway.

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Thanks. You are only adressing in the last part of your answer my original question. The rest of your ellaboration actually adresses a question, which was implicit or even only in the title;) So thanks for reading inbetween the lines. After your explanation, I think the answer is contained in Laumon "Cohomology of Drinfeld modules" Part 1, Proposition 4.7.1, page 114, at least for unramfied extensions. So you claim it is sufficient to prove every thing for the unit? This uses invariance here? I probably have to also include weights in my orbital integrals.... – Marc Palm Nov 26 2011 at 10:19
so I will have to reprove most of the chapter 4 in Laumon. Also my field extensions will be ramified in general. Are there some standard tools, which allow unramified extensions to be treated as ramified ones? – Marc Palm Nov 26 2011 at 10:25

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I am not an expert so I can only offer you a superficial overview rather than practical help with details. My general impression is that in certain situations (e.g. an unramified endoscopic situation), the transfer map is staring at you in the face -- it's coming from the theory of the Satake isomorphism. The problem is proving that the "obvious" transfer does the job you want, i.e. that one orbital integral equals another. I think that if you can check it for the identity functions then you can check it for all the functions, and the fundamental lemma is the assertion that... – Kevin Buzzard Nov 26 2011 at 10:49

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...it's OK for the identity function. So in some simple situations the problem of transferring functions is equivalent to the fundamental lemma. The moment one moves away from these simple situations, I know nothing and I'm afraid I'm not the person to ask. I'll try to remember to take a look at Laumon's book on Monday when I'm back at work, but I am a little worried that it will still mean very little to me on anything other than a superficial level. Ask me the same questions again in 5 years and we'll see if I've learnt any more! I'm still struggling through Labesse-Langlands! – Kevin Buzzard Nov 26 2011 at 10:52
Ok I finally remembered to look. It seems to me that Laumon is just doing what Arthur-Clozel do (but perhaps in char $p$ rather than char 0): proving the base change fundamental lemma. In particular he is proving that an orbital integral downstairs is equal to a twisted orbital integral upstairs. You do not seem to have the twist in your question, so it seems to me that Laumon's calculations are not directly relevant. – Kevin Buzzard Nov 29 2011 at 10:45
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