Zonal Spherical Function - Further Directions

Further Directions

• The theory of zonal functions that are not necessarily positive-definite. These are given by the same formulas as above, but without restrictions on the complex parameter s or ρ. They correspond to non-unitary representations.
• Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions. This is an important special case of his Plancherel theorem for real semisimple Lie groups.
• The structure of the Hecke algebra. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between C_c∞(K \ G / K ) and C_c∞(A)W, the subalgebra invariant under the Weyl group. This is straightforward to establish for SL(2,R).
• Spherical functions for Euclidean motion groups and compact Lie groups.
• Spherical functions for p-adic Lie groups. These were studied in depth by Satake and Macdonald. Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the Langlands program.