Laguerre polynomials, restriction principle, and holomorphic representations of SL(2, ℝ)
The restriction principle is used to implement a realization of the holomorphic representations of SL(2, ℝ) on L2 (ℝ+, tα dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character θ ↔ e-i(2n+α+1)θ under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.
Publication Source (Journal or Book title)
Acta Applicandae Mathematicae
Davidson, M., Ólafsson, G., & Zhang, G. (2002). Laguerre polynomials, restriction principle, and holomorphic representations of SL(2, ℝ). Acta Applicandae Mathematicae, 71 (3), 261-277. https://doi.org/10.1023/A:1015283100541