For real reductive groups 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
that is usually called the
tempered dual
. 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
of
.
We count the maximal tori
of
modulo conjugacy, and we
consider the dual
modulo the action of a certain finite group
called the Weyl group of
.
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 .
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 itself
remains unsolved, except for the groups
where
or
(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 .