Gelfand pair

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Description:
In mathematics, the expression Gelfand pair is a pair (G, K) consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical function in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.

When G is a finite group the simplest definition is, roughly speaking, that the (K,K)-double cosets in G commute. More precisely, the Hecke algebra, the algebra of functions on G that are invariant under translation on either side by K, should be commutative for the convolution on G.

In general, the definition of Gelfand pair is roughly that the restriction to H of any irreducible representation of G contains the trivial representation of H with multiplicity no more than 1. In each case one should specify the class of considered representations and the meaning of contains.

Definitions

In each area, the class of representations and the definition of containment for representations is slightly different. Explicit definitions in several such cases are given here.

Finite group case

When G is a finite group the following are equivalent

  • (G,K) is a Gelfand pair.
  • The algebra of (K,K)-double invariant functions on G with multiplication defined by convolution is......
  • ...

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