Fourier Analysis on Semisimple Symmetric Spaces

A homogeneous space X = G/H of a connected Lie group G is called a symmetric homogeneous space if there exists an involution σ of G such that H lies between the fixed point group G and its identity component Go . Example 0. For a connected Lie group G′, put G = G′×G′, σ(g1, g2) ) = (g2, g1) and H = G. Then the homogeneous space X = G/H is naturally isomorphic to G′ by the map (g1, g2) 7→ g1g−1 2 . Then the action of G on X corresponds to the left and right translations on G′ by elements of G′. Hence any connected Lie group is an example of symmetric homogeneous space. If G′ is the abelian group R in Example 0, then the ring D(R) of invariant differential operators on R equals the ring of differential operators with constant coefficients and L(R) is naturally unitary representation space of R. The irreducible decomposition of the representation is given by Fourier transformations and it is also regards as a spectral resolution of D(R) or expansions of functions in L(R) by joint eigenfunctions of D(R). Considering the above, we give a method of Fourier analysis on X when G is semisimple. Hereafter we assume G is semisimple and first cite more examples.