Gelfand Pair - Symmetric Pairs

Symmetric Pairs

The Gelfand property is often satisfied by symmetric pairs.

A pair (G,K) is called a symmetric pair if there exists an involutive automorphism θ of G such that K is a union of connected components of the group of θ-invariant elements: Gθ.

If G is a connected reductive group over R and K=Gθ is a compact subgroup then (G,K) is a Gelfand pair. Example: G = GL(n,R) and K = O(n,R), the subgroup of orthogonal matrices.

In general, it is an interesting question when a symmetric pair of a reductive group over a local field has the Gelfand property. For symmetric pairs of rank one this question was 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.