Canonical representations on the two-sheeted hyperboloid

A two-sheeted hyperboloid in ℝ3 can be identified with the Riemann sphere ℂ¯, without the equator. Canonical representations of the group SU(1, 1) on ℂ¯ are defined as the restriction of spherical representations of the “overgroup” SL(2, (ℂ¯). We decompose these canonical representations into irreducible constituents and decompose Berezin forms.