The orbit method and the Virasoro extension of Diff⁺ (S¹). I. Orbital integrals

Using the finite dimensional example of PSU(1, 1), the universal covering of PSU(1, 1), as a guide, we revisit the orbit method as it applies to D̂, the universal central extension of D = Diff⁺ (S¹). We clarify some aspects of the classification of coadjoint orbits, determine boundedness properties of the natural height function on these orbits, and calculate orbital integrals.