The Intertwining Lemma

Abstract

This chapter gives a proof of the Intertwining Lemma. Section 20.2 lists out the formulas for all the maps involved. Section 20.3 recalls the definition of Z* and proves Statement 3 of the Intertwining Lemma. Section 20.4 proves statements 1 and 2 of the Intertwining Lemma for a single point. Section 20.5 decomposes Z* into two smaller pieces as a prelude to giving the inductive step in the proof. Section 20.6 proves the following induction step: If the Intertwining Lemma is true for g ɛ GA then it is also true for g + dTA (0, 1). Section 20.7 explains what needs to be done to finish the proof of the Intertwining Lemma. Section 20.8 proves the Intertwining Theorem for points in Π‎A corresponding to the points gn = (n + 1/2)(1 + A, 1 − A) for n = 0, 1, 2, ... which all belong to GA. This result combines with the induction step to finish the proof, as explained in Section 20.7.