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