Harmonic analysis on real reductive symmetric spaces

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 $\mathbb{R}$. 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 $L^2(G\mathrm{/}H)$ with the help of joint eigenfunctions under the algebra of left invariant differential operators under $G$ on $G\mathrm{/}H$.