The classical Hankel transform in the Kirillov model of the discrete series

Abstract

In this paper, we give a new and simple proof of the Hankel inversion formula for the classical Hankel transform of index ν which holds for Re(ν)>−1. Using the proof of this formula, we obtain the full description of the Kirillov model for discrete series representations of SL(2, ℝ) and GL(2, ℝ).

Keywords:

• Bessel functions
• Hankel transform
• Kirillov model
• discrete series

1991 Mathematics Subject Classification::

• Primary: 22E50
• Secondary: 22E35,11S37

Acknowledgements

This research was partially supported by the Center of Excellence of the Israel Science Foundation (grant no. 1691/10).