Abstract
Let G/H be a semisimple symmetric space. Since H is reductive it follows from Helgason [n] Chapter 1, Theorem 1.9 that G/H has an invariant measure, unique up to scalars. Hence the Hilbert space L2 (G/H) makes sense, and we can study the unitary representation
(g,x ∈ G) of G on this space. It is the purpose of L2-harmonic analysis on G/H to give an explicit decomposition (in general as a direct integral) of this representation into irreducibles. So far this program has not been accomplished in general (although the 2 answer is known in several specific cases, notably those of L2 (G/K) and L2 (G × G/d(g)) ≅ L2(G), by the work of Harish-Chandra — see the notes at the end of this chapter).
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© 1984 Birkhäuser Boston, Inc.
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Schlichtkrull, H. (1984). Construction of functions with integrable square. In: Hyperfunctions and Harmonic Analysis on Symmetric Spaces. Progress in Mathematics, vol 49. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-5298-6_8
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DOI: https://doi.org/10.1007/978-1-4612-5298-6_8
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-9775-8
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