  Let $G$ be a reductive group in the Harish-Chandra class e.g. a connected
semisimple Lie group with finite center, or the group of real points of a
connected reductive algebraic group defined over $\R$. Let $\sigma$ be an
involution of the Lie group $G$, $H$ an open subgroup of the subgroup of fixed
points of $\sigma$. One decomposes the elements of $L^2(G/H)$ with the help of
joint eigenfunctions under the algebra of left invariant differential operators
under $G$ on $G/H$.
