17 - The discrete series of S1₂(ℝ)

Summary

Let (π,H) be a unitary irreducible representation in the discrete series of some group G. Fix u ∈H with ∥u∥² = d and consider the coefficients c_v = c^u_v of π. By definition of the discrete series, they are square summable on G and more precisely (16.3.a) shows that

is an isometry which embeds π in the right regular representation r of G. If a sequence v_n → v in H, → c_v uniformly on G

In the model of π consisting of right translations in {c_v : v ∈ H}, all functions c_v are continuous and

→ c_v in L² (G) ⇒ → c_v uniformly

(because the assumption is equivalent to v_n → v in H). This situation is very peculiar and reminds one of properties of holomorphic (or harmonic) functions. Indeed, the discrete series of Sl₂(ℝ) will be constructed in spaces of holomorphic (or anti-holomorphic) functions.

The group Sl₂(ℝ) acts on the upper half-plane Im(z) > 0, but it will be more convenient to let it act in the unit disc |w| < 1 (conformally equivalent to the upper half-plane) and thus, to use a conjugate SU(1,1) of S1²(ℝ) in S1₂(ℂ). Here is a description of this new group.