Fourier analysis on domains in compact groups

Let Ω be a measurable subset of a compact group G of positive Haar measure. Let be a non-negative function defined on the dual space and let L2(μ) be the corresponding Hilbert space which consists of elements (ξπ)π∈suppμ satisfying , where ξπ is a linear operator on the representation space of π, and is equipped with the inner product: . We show that the Fourier transform gives an isometric isomorphism from L2(Ω) onto L2(μ) if and only if the restrictions to Ω of all matrix coordinate functions , π∈suppμ, constitute an orthonormal basis for L2(Ω). Finally compact connected Lie groups case is studied.