Multiplicative Hitchin Fibrations and the Fundamental Lemma

Abstract

Let $k$ be a finite field and let $G$ be a reductive group over $k[[\pi ]]$. Suppose $\mathrm{char}(k)$ is larger than twice the Coxeter number of $G$, we prove the standard endoscopic fundamental lemma for the spherical Hecke algebra of $G$ using multiplicative Hitchin fibrations.

arXiv comment265 pages. Accepted, to appear as an Annals of Math Studies volume. Revision addressing referees' comments: (1) substantially restructure and rewrite part of chapter 5, especially the current 5.2; (2) upgrade the product formula in 6.9 using results from newer literature; (3) numerous small fixes and adjustments suggested by the referees; (4) add an index and list of notations

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