an equvalent defintion of Gelfand pair is equvalent is that the G - representetion F(G/K) is multiplicity free, where F(G/K) is some function space on G/K. Again in each case one should specify the class of functions and the meaning of "multiplicity free". The equvalence of the definition folows from Frobenius reciprocity

Operetions with Gelfand pairs

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Let   and   be gelfand pairs then (generaly spiking) the pair   is a gelfand pair.

If (G, K) is a Gelfand pair, then (G/N, K/N) is a Gelfand pair for every G-normal subgroup N of K. More genraly, for any normal subgroup N of G the pair   is a Gelfand pair.

Harmonic analisis

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If (G, K) is a Gelfand pair it means that the space of functions on G/K "decomposes" to distinct irredusebale representations of G. Now assum that we know the clasification of irredusebale representations of G and we have a fixed basis for eich, then we can obtain a basis for the space of functions on G/K, in wich it will be convinent to consider the action of G. For example if we will consider in such a way the pair (SO_3, SO_2) we wil gat the theory of spherical harmonics