Limits of Discrete Series Abstract
### Anthony W. Knapp

** Abstract of "Limits of Discrete Series with Infinitesimal Character Zero"
**

For a connected linear semisimple Lie group G, this paper considers those
nonzero limits of discrete
series representations having infinitesimal character 0, calling them
* totally degenerate. *
Such representations exist if and only if
G has a compact Cartan subgroup, is quasisplit, and is acceptable in
the sense of Harish-Chandra.

Totally degenerate limits of discrete series are natural objects of study
in the theory of automorphic forms: in fact, those automorphic
representations of adelic groups that have totally degenerate limits of
discrete series as archimedean components correspond conjecturally to
complex continuous representations of Galois groups of number fields. The
automorphic representations in question
have important arithmetic significance, but very little has been proved up
to now toward establishing this part of the Langlands conjectures.

There is some hope of making progress in this area, and for that one
needs to know in detail the representations of G under consideration.
The aim of this paper is to determine the classification parameters of
all totally degenerate limits of discrete series in the Knapp-Zuckerman
classification of irreducible tempered representations, i.e., to express
these representations as induced representations with
* nondegenerate data.*

The paper uses a general argument, based on the finite
abelian reducibility group
R attached to a specific unitary principal series representation of G.
First an easy result gives the aggregate of the classification
parameters. Then a harder result uses the easy result to match the
classification parameters with the representations of G under consideration
in representation-by-representation fashion. The paper includes tables of the
classification parameters for all such groups
G.