Probability and Mathematical Statistics: Vol. 33, Fasc. 2

CLASSICAL METHOD OF CONSTRUCTING A COMPLETE SET OF IRREDUCIBLE REPRESENTATIONS OF SEMIDIRECT PRODUCT OF A COMPACT GROUP WITH A FINITE GROUP

Takeshi Hirai

Abstract: Let $G = U ⋊ S$ 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 $^ [ρ] ∈ U$ . Intertwining operators $Jρ(s)(s ∈ S ([ρ]))$ between $ρ$ and $s-1 ρ$ gives in general a spin (= projective) representation of $S([ρ])$ , which is lifted up to a linear representation $′ Jρ$ of a covering group $′ S ([ρ])$ of $S([ρ])$ . Put $0 ′ π := ρ⋅Jρ$ , and take a spin representation $1 π$ of $S([ρ])$ corresponding to the factor set inverse to that of $Jρ$ , and put $0 1 G 0 1 Π (π ,π ) = IndU⋊S([ρ])(π ⊡ π )$ . We give a simple proof that $0 1 Π (π ,π )$ is irreducible and that any IR of $G$ is equivalent to some of $0 1 Π(π ,π )$ .

2000 AMS Mathematics Subject Classification: Primary: 20C99; Secondary: 20C15, 20C25.

Keywords and phrases: Semidirect product group, construction of irreducible representations, projective representation, finite group and compact group.