User:Aizenr/gelfand pair

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
Let $$(G_1,K_1)$$ and $$(G_2,K_2)$$ be gelfand pairs then (generaly spiking) the pair $$(G_1 \times G_2,K_1 \times K_2)$$ 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 $$(G/N, K/(N \cap K))$$ is a Gelfand pair.

Harmonic analisis
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