Let G be a semisimple Lie
group and let H be a closed reductive subgroup of G. The homogeneous space
X = G∕H is called a semisimple homogeneous space. A fundamental goal of
harmonic analysis is to understand the group action of G on the various function
spaces of X. In particular, the L2-harmonic analysis on X is to decompose
L2(X) as a direct integral of irreducibles, i.e., to find a family of irreducible
unitary representations {Vω∣ω ∈ Ω} of G, and a measure ν on the set Ω, so
that
(1)
The above decomposition is called the Plancherel formula for the homogeneous space
X. In this paper we prove the Plancherel formulae for some non-symmetric
semisimple homogenous spaces.
