Harish-Chandra's Plancherel formula

For real reductive groups $G(\mathbb R)$ the Plancherel formula was found by Harish-Chandra, in 1976, after a monumental effort that took him more than 20 years. What is described by Harish-Chandra is the support of the Plancherel measure, the part of $\hat G$ that is usually called the tempered dual $\hat G^{temp}$. What comes out of the analysis of Harish-Chandra is, roughly, that the tempered representations arise in series of various dimensions that are parametrized by the dual groups of the maximal tori $T$ of $G(\mathbb R)$. We count the maximal tori $T$ of $G(\mathbb R)$ modulo conjugacy, and we consider the dual $\hat T$ modulo the action of a certain finite group called the Weyl group of $T$.

The Plancherel formula of Harish-Chandra is a very beautiful and complex result. To obtain this result it was necessary to introduce numerous new notions. One of the cornerstones is the construction of the so called discrete series. These representations can be viewed as the basic building blocks for general tempered representations. Harish-Chandra has given a complete classification of the discrete series representations of any real reductive group $G(\mathbb R)$.

The construction of tempered representations of real reductive groups is reasonably well understood at present. Modern constructions are based on geometric (cohomological) methods. The precise classification of the irreducible tempered representations requires a complicated and detailed study of reducibility questions for singular induced representations. This problem was solved by Knapp and Zuckerman in 1977.

However, the question to describe $\hat G$ itself remains unsolved, except for the groups $GL(n,F)$ where $F=\mathbb R$ or $\mathbb C$ (Vogan, 1986) and for a few examples of groups of small rank. One can classify the larger class of so-called admissible irreducible representations, but the problem to decide if a given admissible irreducible representation is unitarizable is out of reach at present. This is considered to be a very important open problem.

After finishing the proof of the Plancherel formula for real reductive groups, Harish-Chandra turned his attention to the study of p-adic reductive groups. He was able to derive a Plancherel formula in this case as well, but he was not able to classify the discrete series representations in this case. This classification problem is still open today, except for the case of $GL(n,F)$.