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.
Comment: 16 pages, no figures, contribution to GROUP33/35 held in Cotonou, Benin, July 15 - 19, 2024