In his paper on the p-adic analogue of the Kazhdan-Lusztig hypothesis (Functional Analysis and Its Applications 15.2 (1981): 83-92), Zelevinskii proves a combinatorial proposition (outlined in Section 2.6 of the paper) using a lemma that he calls the "Fundamental Lemma". Is this lemma, by the power of some brilliant analogy, similar to what is called the "Fundamental Lemma" in the Langlands program (recently proved by Ngo et al) ? Or, is it just a case of an accidental match of terminology ?
1 Answer
There are many fundamental lemmas, not only Langland's fundamental lemma (with a proof by Ngo) in the theory of automorphic forms. Of course, this one is a particularly deep result. I do not really see an analogy to the result of Zelevinskii.
There are other "fundamental lemmas" in the theory of automorphic forms, namely the
stable base change fundamental lemma of Clozel, or the Jacquet-Ye relative fundamental lemma. Here there are relations.
Other fundamental lemmas are to be found in homological algebra, in geometric group theory
(by Svarc and Milnor), in the calculus of variations, etc.
For the question of an example, see Bill Casselmann's article "Langlands’ Fundamental Lemma for $SL_2$.
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$\begingroup$ Thanks. Is there some pedagogical account of the stable base change lemma ? Say, with some easy to prove examples etc. As for my question, I will wait to see if anybody else has something to say. Any analogy between the two cases would be helpful for me since I have some feel for what Zelevinsky is trying to do in that paper. But, any treatment of the Langlands-Shelstad Fundamental Lemma quickly becomes quite involved. $\endgroup$ Oct 18, 2013 at 14:05