Harmonic analysis on real reductive symmetric spaces

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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. Let σ be an involution of the Lie group G, H an open subgroup of the subgroup of fixed points of σ. One decomposes the elements of L²(G/H) with the help of joint eigenfunctions under the algebra of left invariant differential operators under G on G/H.