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 ΔKG × 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.

Read more about this topic:  Gelfand Pair

Famous quotes containing the word strong:

    The weak are the most treacherous of us all. They come to the strong and drain them. They are bottomless. They are insatiable. They are always parched and always bitter. They are everyone’s concern and like vampires they suck our life’s blood.
    Bette Davis (1908–1989)