Analysing the Schmid Operator for SL(2,R) and SU(2,1)
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Physics (MPPHS), MSc
Publicerad
2023
Författare
Olander, Oskar
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
The study of holomorphic discrete series on the special linear group SL(2, R)
has been very fruitful in many areas in mathematics and physics, so there is a
natural question to ask if they can be generalised to other Lie groups. There
exists a differential operator called the Schmid operator which can be used to
define discrete series representations of semisimple Lie groups. Understanding
the Schmid operator is one of the main goals of this thesis. A second goal is
to describe the Schmid operator in a tensorial formalism which is more closely
related to differential geometry. We show that the Schmid operator before the
projection is equivalent to a covariant derivative in the tensorial formalism. The
Schmid operator is given explicitly for the Lie groups SL(2, R) and SU(2, 1). In
the case of SL(2, R) the conditions for holomorphic and anti-holomorphic discrete
series are retrieved. In the case of SU(2, 1) the conditions for holomorphic, antiholomorphic, and quaternionic discrete series are retrieved.
Beskrivning
Ämne/nyckelord
Schmid operator, discrete series representation, holomorphic discrete series, quaternionic discrete series, symmetric space