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:

    Learning is, in too many cases, but a foil to common sense; a substitute for true knowledge. Books are less often made use of as “spectacles” to look at nature with, than as blinds to keep out its strong light and shifting scenery from weak eyes and indolent dispositions.... The learned are mere literary drudges.
    William Hazlitt (1778–1830)