- k)× Sym(k), Sym(n/2) wr Sym(2), Sym(2) wr Sym(n/2) for n even, Sym(n - 5) × AGL(1,5), Sym(n - 6) × PGL(2,5), or Sym(n - 9) × PΓL(2,8). Gelfand pairs for classical groups have been investigated as well. Symmetric pairs with compact K (GL(n,R), O(n,R)) (GL(n,C), U(n)) (O(n+k,R), O(n,R) × O(k,R)) (U(n+k), U(n) × U(k)) (G,K) where G is a reductive Lie group and K is a maximal compact subgroup. Symmetric Gelfand pairs of rank one Let F be a local field of characteristic zero. (SL(n+1,F), GL(n,F)) for n > 5. (Sp(2n+2,F), Sp(2n,F)) × Sp(2,F)) for n > 4. (SO(V ⊕ F), SO(V)) where V is a vector space over F with a non-degenerate quadratic form. Symmetric pairs of high rank Let F be a local field of characteristic zero. Let G be a reductive group over F. The following are examples of symmetric Gelfand pairs of high rank: (G × G, ΔG): Follows from Schur's lemma. (GL(n+k,F), GL(n,F) × GL(k,F)). (GL(2n,F), Sp(2n,F)). (O(n+k,C), O(n,C) × O(k,C)). (GL(n,C), O(n,C)). (GL(n,E), GL(n,F)), where E is a quadratic extension of F. Strong Gelfand pairs The following pairs are strong Gelfand pairs: (Sym(n+1), Sym(n)), this is proven using the involutive anti-automorphism g ↦ g−1. (GL(n+1,F), GL(n,F)) where F is a local field of characteristic zero. (O(V ⊕ F), O(V)) where V is a vector space over F with a non-degenerate quadratic form. U(V ⊕ E), U(V)) where E is a quadratic extension of F and V is a vector space over E with a non-degenerate hermitian form. Those four examples can be rephrased as the statement that the following are Gelfand pairs: (Sym(n+1) × Sym(n), Δ Sym(n)). (GL(n+1,F) × GL(n,F), Δ GL(n,F)) (O(V ⊕ F) × O(V), Δ O(V)) (U(V ⊕ E) × U(V), Δ U(V)) See also spherical function Symmetric pair Spherical pair Notes References Representation theory of groups Representation theory of Lie groups Harmonic analysis