The SL(2,R) Group Representations on Spaces of Holomorphic Functions on the Unit Disc
DOI:
https://doi.org/10.29020/nybg.ejpam.v16i4.4923Keywords:
Representation, SL_2(R), SU(1 .1), Dirichlet spaceAbstract
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.
Downloads
Published
Issue
Section
License
Copyright (c) 2023 European Journal of Pure and Applied Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Upon acceptance of an article by the European Journal of Pure and Applied Mathematics, the author(s) retain the copyright to the article. However, by submitting your work, you agree that the article will be published under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). This license allows others to copy, distribute, and adapt your work, provided proper attribution is given to the original author(s) and source. However, the work cannot be used for commercial purposes.
By agreeing to this statement, you acknowledge that:
- You retain full copyright over your work.
- The European Journal of Pure and Applied Mathematics will publish your work under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).
- This license allows others to use and share your work for non-commercial purposes, provided they give appropriate credit to the original author(s) and source.