Plancherel theorem for spherical functions | Wikipedia audio article

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00:01:14 1 History
00:04:37 2 Spherical functions
00:09:49 3 Spherical principal series
00:11:19 4 Example: SL(2,C)
00:11:26 5 Example: SL(2,R)
00:11:37 5.1 Hadamard's method of descent
00:25:06 5.2 Flensted–Jensen's method of descent
00:29:03 5.3 Abel's integral equation
00:47:14 6 Other special cases
01:05:39 6.1 Complex semisimple Lie groups
01:17:23 6.2 Real semisimple Lie groups
01:18:38 7 Harish-Chandra's Plancherel theorem
01:30:36 8 Harish-Chandra's spherical function expansion
01:42:44 9 Harish-Chandra's c-function
01:52:59 10 Paley–Wiener theorem
02:05:22 11 Rosenberg's proof of inversion formula
02:15:54 12 Schwartz functions
02:17:25 13 Notes
02:21:30 14 References



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SUMMARY
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In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations.
It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of
hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.
The main reference for almost all this material is the encyclopedic text of Helgason (1984).