The SL(2,R) Group Representations on Spaces of Holomorphic Functions on the Unit Disc

Abstract

We can realise the representations of the group SL(2,R) on the unit disc. This is due to an isomorphism between the group SL(2,R) and the group SU(1,1). The discrete series representations for the group SL(2,R)given by

\pi_{n}(g)\varphi(z)=\varphi (\frac{d z-b}{a-cz} )(a-c z)^{-n},    where  n  is an integer number,
is on the Bergman space where n>2 .
Lang studies the discrete series on the group  in the upper half-plane and on the unit disc.  For n=1, the SL(2,R) representation is called the mock discrete series. The representation space of the mock discrete series is the Hardy space.
In this article we describe the  SL(2,R) representation on the Dirichlet space.