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Classical method of constructing a complete set of irreducible representations of semidirect product of a compact group with a finite group

Hirai T.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

353--362

Let G = U x S be a group of semidirect product of U compact and S finite. For an irreducible representation (= IR) ρ of U, let S([ρ]) be the stationary subgroup in S of the equivalence class [ρ] ∈ Û. Intertwining operators Jρ(s) (s ∈ S([ρ])) between ρ and s−1ρ gives in general a spin (= projective) representation of S([ρ]), which is lifted up to a linear representation J′ρ of a covering group S([ρ])′ of S([ρ]). Put π0:= ρ ·J′ρ, and take a spin representation π1 of S([ρ]) corresponding to the factor set inverse to that of Jρ, and put Π(π0, π1) = IndGU x S([ρ]) (π0 ▪ π1). We give a simple proof that Π(π0, π1) is irreducible and that any IR of G is equivalent to some of Π(π0, π1).