Gelfand Pair - Strong Gelfand Pairs

Strong Gelfand Pairs

A pair (G,K) is called a strong Gelfand pair if the pair (G × K, ΔK) is a Gelfand pair, where ΔK ≤ G × K is the diagonal subgroup: {(k,k) in G × K : k in K}. Sometimes, this property is also called the multiplicity one property.

In each of the above cases can be adapted to strong Gelfand pairs. For example, let G be a finite group. Then the following are equivalent.

  • (G,K) is a strong Gelfand pair.
  • The algebra of functions on G invariant with respect to conjugation by K (with multiplication defined by convolution) is commutative.
  • For any irreducible representation π of G and τ of K, the space HomK(τ,π) is no more than 1-dimensional.
  • For any irreducible representation π of G and τ of K, the space HomK(π,τ) is no more than 1-dimensional.

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