Fundamental lemma (twisted case)

Abstract

In this chapter we show under conditions formulated in Sect. 9.2 that the fundamental lemma for standard endoscopy implies the fundamental lemma for twisted base change endoscopy. In fact, this is shown in this chapter for the unit elements of the Hecke algebras.Using global arguments involving the Selberg trace formula, it suffices to prove the fundamental lemma and twisted fundamental lemma in general assuming the fundamental lemma for unit elements and standard endoscopy for residue characteristic p ≥ p₀. See Corollary 9.4 and Chap. 10. It also implies the Frobenius formula (see Lemma 9.7) used in the comparison of trace formulas in Chap. 3. This formula will be the clue to unravel the terms in the Kottwitz formula stated in Theorem 3.1 that appear in the form of the twisted orbital integrals TO_δ(φ_n).