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(2) 22. Mf. No.44. t. Two multipliersa, a' are called equivalentmutipliers if there is a Borel map p : G —>T such that. Hence the unitary operator 11(g) gives an intertwining operator between 7rg and 7r for each. al(91,92) =0-(91,92)P(91)P(92)g1, g2 E G. P(9192). The rest of this paper let {7r, RI be a unitary representation of N, and let. Throuout of this paper, G denotes a separable locally compact group and N denotes a closed normal subgroupof G. On locally compact groups,it holds that G is separable if G is second coutable. Let {7r,1-1}be a unitary representation of N. Then G acts naturally on the space of unitary representations of N as follows: 7rg(n):= r(g—ing), n E N, g E G. If 7r is irreducible, this action works on the dual N of N. Suppose that if there exists a projective representation ft of G which is an extension of 7r. The equivalence class of mutipliers on G which are equivalent to the mutiplier of II is known as the Mackey obstruction to extending the representation 7rof N to G. In circumstances described above, it follows immediately that Lemma. n can be taken as unitary iff Mackey obstruction is trivial, that is, trivial mutiplier 1 belongsto it. 3. Unitary Extensions In order to state and to prove our Theorem, we prepare some Propositions.. g E G. Thus the assertion follows.. {ir,. = j {rr., 'Hy}dµ(-y) Jr. ^. (1). be an irreducible decompositin of far, 70. Let denote the diagonal algebra by A which is isomorphic to L'(1-1, 11). Proposition 2. Let assume that rg is unitarily equivalentto 7rfor each g E G, and let Wg be an intertwining operator between7r9and 7r. Then it follows that Wg is decomposableif Wg belongs to 931(7r).. Proof.Sincex = f®xydp commutes with ir(N) if zy E lry(N)' for a.e. -yE F, it follows that OW' = 7r(N)' n A', where M' denots the commutant of M. Hence we have that 931(7r)is generated by 7r(N) and A. Since the decomposition (1) is irreducible, the diagonal algebra A is a maximal abelian subalgebra of 7r(N)', so that A' = 9X(7r). Since Wg is decomposable if Wg E A', consequently it followsthat Wg is decomposable if Wg belongs to 9jt(7r). ^ Proposition 3. In the decomposition(1), if for each g E G, 7r?),is unitarily equivalentto iry a. e. 7 E F, then rg is unitarily equivalentto ir foe each g E G.. Proposition 1. Let {7r,10 be a unitary representation of N. Suppose that 7r extends to a unitary representation {11,1-1}of G, then it holds that ergis unitarily equivalentto ir for each g E G. Proof. Since H is an extension of 7r, for each g E G, n E N it holds that irg(n) = 7r(g-1ng) = H(g—ing). Proof. Let denote Wg an intertwiningoperator between el, and 'myfor each g E G, a.e. -y on the space 711,. Then for each g E G, n E N, it holds that. /1"yg(n) = (W,.)*. (2). Since 3 ry 1-4 E WW() is a Borelfieldof unitary operators,for eachg E G let define. Wg frdyey).. = 11(g-1)11(n)H(g) = H (g)-17r(n)11(g) = Mg)* 7r(n)11(g).. Wyg , a.e.7 E. Then. Wg. space. 'H.. is a unitary. operator. on the. (3) Hilbert.
(3) 23. Now we Now we are are in in the the position to state and to prove our prove our main main result. result. ( cf. Corollary V.8,t21). Since it follows from (1) that. {Kg, RI=jr,{-71-9,11-ydy(7), gEG, (4) Theorem. it followsfrom (1),(2),(3),(4) that the unitary operator wg satisfies the followingequation,. Let 7r be a unitary representation of a closed normal subgroupN of G. Then ir extends to a unitary representation of G if for each g E G the representation rg is unitarily equivalent. to 7r and there exists an intertwining operator beirg(n) =f7q,(n)dic(nt) longing to MOO and the Mackey obstruction is =f(W,6,)*7r,y(n)1471 ,gdp,(-y) trivial.. =(f. Proof. From Proposition 4, it followsthat dµ(y) for each g E G, 7rgyis unitarily equivalent to iny for a.e. -yE F. Hence there exists a projective. is 7r7(n)dp(-y) fr Wg. = (Wg)*7r(n)Wg, n E N,g E G. Hence we have that for each g E G, the unitary operator Wg gives an intertwining operator between 7rgand 7T.Thus the assertion follows. 0 Proposition 4. In the decomposition(1), if irg is unitarily equivalentto ir for eachg E G, and there exists an intertwining operator belongingto 931(7r),then 7r,Fy is unitarily equivalent to Tryfor a.e. -yE F. Proof. Let denote by sg an intertwining operator between 7rg and 7r. Then we have for each g E G, irg(n) = (Sg)*7r(n)Sg, n, E N.. (5). By the assumption we can take Sg from 9347r)so that the unitary operator Sg is decomposablefor each g E G by Proposition 2. So we have sg =. S,yg dp(y).. (6). representation 14 of G suchthat fry(n) = 7r-y(n), n E N, a.e.-yE r. (7). by virtureofthe Mackey's originalresult.Hence if Mackeyobstructionistrivial,wegeta unitary representation 17yofG in placeoffry fora.e.y E F. Sincethe representation space7-i of Hi, is the selfsameas (7r,y)'s for a.e.7 E F, the map 111,is Borel. Indeedlet al, E 117be a non-zerovectorfor a.e.y E r and g E G,n E N, E Ity.. Since {7r1„1-t-Jis irreducible, non-zero vector ay is a cyclicvector fora.e. -y E F. Hence for each k E N, there exist n2,nj E N and ci, di E C such that. (Thy(g) ,),177.1) TEeidien-7(n21ni)9710-01 < —l c. This. implies. tions. of G;. that. the field of unitary y 1-4 al,. representa-. is Borel.. Let define. By (1),(5),(6) it followsfor each n E N, g E G. 9-11 :=f{Thy, 74-Jdp(-y).. -r fre) 7r (n) dp(-y) =f (S,g1")* 7ri,(n)S4 cittey).Since it followsfrom (7) that Hence we have for each g E G, 7r,rg(n)= (Sgy)*7r-y(n)Sgr n E N, a.e.-yE F. This implies that for each g E G the unitary operator SF),on n.), gives an intertwining operator between ey and Tryfor a.e.ey E F. Thus the assertion follows. ^. r. r. (i). 17(n)= J 17y(n) dp(y). I177(n) dµ(y) =. 7ri,(n)dµ(-y). = 7r(n), n E N.
(4) No.44. 24. , we conclude that a unitary representation ir of N extends to a unitary representation 11 of G. This completes. the proof.. ^. References. [1]Duflo,M.: Sur les extensions des representations irreductibles des groupes de Lie nilpotents, Ann.Ec,Norm.Sup.,71-120,(1972) [2]Fabec,R.C. : Fundamentals of Infinite Dimensional Representation Theory, Chapmn,Hall/Crc (2000) [3]Huef,A.,Kaliszewski,S.,Raeburn,I.,Extending representations of subgroups and the duality of induction and restriction, arXiv:math.OA/0312283 vl, (2003) [4]Macky,G.W.: Unitary representations of group extensions, I , Acta Math.,(99), 265-311, (1958).
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