Gelfand Pair - Spherical Pairs

Spherical Pairs

If G is a reductive group over a local field there is another property that is weaker than the Gelfand property, but is easier to verify. Namely, the pair (G,K) is called a spherical pair if one the following equivalent conditions holds.

• For any parabolic subgroup P of G there exists an open (P,K)-double coset in G.
• For any parabolic subgroup P of G there is a finite number of (P,K)-double cosets in G.
• For any admissible representation π of G, the space Hom_K(π,C) is finite dimensional.