Notes on the Knapp-Zuckerman theory

The point of these notes is to redefine some of their concepts in terms of the L-group. I observe, however, that it is best and indeed essential for further applications that their results be formulated for reductive groups rather than just for simply-connected semi-simple groups. I use the notation of CIRRAG (On the classification of irreducible representations of real algebraic groups) modified sometimes according to Borel’s suggestions. Since we are dealing with tempered representations we start from φ : WC/R → G with image which is essentially compact. We suppose φ defines an element of Φ(G). Choose a parabolic P in G which is minimal with respect to the property that φ(WC/R) ⊆ P . P defines P and M . Let ρ (with character Θ) be one of the representations of M associated to φ. Thus ρ ∈ Πφ, if φ is regarded as taking WC/R to M . It is