User:Aizenr/sandbox

Sphirical pairs
In the context of algebraic groups the analoges of Gelfand pairs are called spherical pair. Namely, A pair (G,K) of algebraic groups is called spherical pair if one the following equivalent conditions holds.


 * There exists an open (B,K)-double coset in G, where B is the Borel subgroup of G.
 * There is a finite number of (B,K)-double coset in G
 * For any algebraic representation π of G, we have dim π^K \leq 1.

In this case the space G/H is called spherical space.

It is conjectured that any spherical pair (G,K) over a local field, satisfies the following weak version of the Gelfand property: For any admissible representation π of G, the space HomK(π,C) is finite dimensional. Moreover the bound for this dimension dose not depend on π. This conjecture is proven for a large class of spherical pair including all the symmetric pairs.