Construction of functions with integrable square

Abstract

Let G/H be a semisimple symmetric space. Since H is reductive it follows from Helgason [n] Chapter 1, Theorem 1.9 that G/H has an invariant measure, unique up to scalars. Hence the Hilbert space L² (G/H) makes sense, and we can study the unitary representation

$$ \left( {\pi (g)f} \right)\left( {x\;H} \right) = f\left( {{g^{ - 1}}x\;H} \right) $$

(g,x ∈ G) of G on this space. It is the purpose of L²-harmonic analysis on G/H to give an explicit decomposition (in general as a direct integral) of this representation into irreducibles. So far this program has not been accomplished in general (although the 2 answer is known in several specific cases, notably those of L² (G/K) and L² (G × G/d(g)) ≅ L²(G), by the work of Harish-Chandra — see the notes at the end of this chapter).