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Induced representation
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===Geometric=== Suppose {{mvar|G}} is a [[topological group]] and {{mvar|H}} is a [[Closed set|closed]] [[subgroup]] of {{mvar|G}}. Also, suppose {{mvar|π}} is a representation of {{mvar|H}} over the vector space {{math|''V''}}. Then {{mvar|G}} [[Group action (mathematics)|acts]] on the product {{math|''G'' Γ ''V''}} as follows: :<math>g.(g',x)=(gg',x)</math> where {{math|''g''}} and {{math|''g''β²}} are elements of {{mvar|G}} and {{math|''x''}} is an element of {{math|''V''}}. Define on {{math|''G'' Γ ''V''}} the [[equivalence relation]] :<math>(g,x) \sim (gh,\pi(h^{-1})(x)) \text{ for all }h\in H.</math> Denote the equivalence class of <math>(g,x)</math> by <math>[g,x]</math>. Note that this equivalence relation is invariant under the action of {{mvar|G}}; consequently, {{mvar|G}} acts on {{math|(''G'' Γ ''V'')/~}} . The latter is a [[vector bundle]] over the [[Quotient space (topology)|quotient space]] {{math|''G''/''H''}} with {{math|''H''}} as the [[structure group]] and {{math|''V''}} as the fiber. Let {{math|''W''}} be the space of sections <math>\phi : G/H \to (G \times V)/ \! \sim</math> of this vector bundle. This is the vector space underlying the induced representation <math>\operatorname{Ind}_H^G\pi : W \to \mathcal L_W</math>. The group {{mvar|G}} acts on a section <math>\phi : G/H \to (G \times V)/ \! \sim</math> given by <math>gH \mapsto [g,\phi_g]</math> as follows: :<math>(g\cdot \phi)(g'H)=[g',\phi_{g^{-1}g'}] \ \text{ for } g,g'\in G.</math>
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