The Discrete Series for a Semi-Simple Lie Group — Existence and Exhaustion

Abstract

Let G be an acceptable connected semi-simple Lie group with finite center, Ĝ its unitary dual. Let Ĝ _d be the discrete series for G. Given Û ∊ Ĝ _d , let To denote its character, d _Û its formal dimension — then, as is known, the distribution

$${T_{d}} = \sum\limits_{{\hat{U} \in {{\hat{G}}_{d}}}} {{d_{{\hat{U}}}}{T_{{\hat{U}}}}}$$

represents the contribution of the discrete series to the Plancherel formula for G. This being so, the primary objectives of the present chaper are as follows:

• (1) Determine when the set Ĝ _d is non-empty;

• (2) Describe explicitly the entities d _Û , T _Û (Û ∊ Ĝ _d ) and T _d .