Tempered Representation - Classification

Classification

The irreducible tempered representations of a semisimple Lie group were classified by Knapp and Zuckerman (1976, 1982). In fact they classified a more general class of representations called basic representations. If P=MAN is the Langlands decomposition of a cuspidal parabolic subgroup, then a basic representation is defined to be the parabolically induced representation associated to a limit of discrete series representation of M and a unitary representation of the abelian group A. If the limit of discrete series representation is in fact a discrete series representation, then the basic representation is called an induced discrete series representation. Any irreducible tempered representation is a basic representation, and conversely any basic representation is the sum of a finite number of irreducible tempered representations. More precisely, it is a direct sum of 2r irreducible tempered representations indexed by the characters of an elementary abelian group R of order 2r (called the R-group). Any basic representation, and consequently any irreducible tempered representation, is a summand of an induced discrete series representation. However it is not always possible to represent an irreducible tempered representation as an induced discrete series representation, which is why one considers the more general class of basic representations.

So the irreducible tempered representations are just the irreducible basic representations, and can be classified by listing all basic representations and picking out those that are irreducible, in other words those that have trivial R-group.