Plancherel theorem for spherical functions

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 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. In the case of hyperbolic space, these expansions were known through the prior work of Mehler, Weyl and Fock.

References

• Helgason, Sigurdur (1984), Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, ISBN 0-12-338301-3