investigated in and An example of high rank Gelfand symmetric pair is (GL(n+k), GL(n) × GL(k)). This was proven in over non-archimedean local fields and later in for all local fields of characteristic zero. For more details on this question for high rank symmetric pairs see. Spherical pairs In the context of algebraic groups the analogs of Gelfand pairs are called spherical pair. Namely, a pair (G,K) of algebraic groups is called a spherical pair if one of 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 . 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 does not depend on π. This conjecture is proven for a large class of spherical pairs including all the symmetric pairs. Applications Classification Gelfand pairs are often used for classification of irreducible representations in the following way: Let (G,K) be a Gelfand pair. An irreducible representation of G called K-distinguished if HomK(π,C) is 1-dimensional. The representation Ind(C) is a model for all K-distinguished representations i.e. any K-distinguished representation appears there with multiplicity exactly 1. A similar notion exists for twisted Gelfand pairs. Example: If G is a reductive group over a local field and K is its maximal compact subgroup, then K distinguished representations are called spherical, such representations can be classified via the Satake correspondence. The notion of spherical representation is in the basis of the notion of Harish-Chandra module. Example: If G is split reductive group over a local field and K is its maximal unipotent subgroup then the pair (G,K) is twisted Gelfand pair w.r.t. any non-degenerate character ψ (see,). In this case K-distinguished representations are called generic (or non-degenerate) and they are easy to classify. Almost any irreducible representation is generic. The unique (up to scalar) imbedding of a generic representation to Ind(ψ) is called a Whittaker model. In the case of G=GL(n) there is a finer version of the result above, namely there exist a finite sequence of subgroups Ki and characters s.t. (G,Ki) is twisted