A Kac-Moody algebra associated to the non-compact manifold SL$(2,\mathbb{R})$

We construct explicitly a Kac-Moody algebra associated to SL$(2,\mathbb{R})$ in two different but equivalent ways: either by identifying a Hilbert basis of $L^2($SL$(2,\mathbb{R}))$ or by the Plancherel Theorem. Central extensions and Hermitean differential operators are identified.