Applications and examples - UNITARY REPRESENTATIONS OF GROUP EXTENSIONS. I

Suppose t h a t (~, :~ and a are as described in the first sentence of Theorem 8.1 and suppose that the only quasi orbits of :~~ under {~ are transitive ones. Then the primary a representations of (~ described in Theorem 8.4 include all primary a representations of (~

and we reduce the problem of finding the primary representations of (~ to t h a t of finding the orbits of :~" under (~ and for each such orbit to finding the primary co representations of a certain subgroup of (~/:K. Moreover it is easy to find useful sufficient conditions for the absence of non transitive quasi orbits and hence for the possibility of the indicated complete analysis. Let us say t h a t :~ is a regularly imbedded in (~ whenever for each finite Borel measure # in }(" the measure ~ in the orbit space (:~")" is eountably separated. This says slightly less than t h a t (:~")- is metrically countably separated since we do not know t h a t every finite Borel measure in (~(~)~ is of the form ~. Now the hypothesis of metric countable separatedness for S made in the statement of Theorem 6.3 is not used in the proof in its full force. All t h a t is actually used is t h a t measures in S of the form ~are countably separated. Thus we m a y apply Theorems 6.3 and 7.6 and conclude the t r u t h of

THEOREM 9.1. I / (~, ~ and a are as described in the/irst sentence o/Theorem 8.1 and :~

is ~ regularly imbedded in (~ then every quasi orbit o / ~ under ~ is transitive. Hence, in partic- ular, every primary a representation o] (~ is one o/those described in Theorem 8.4.

We have also

THEOREM 9.2. Let (~, ~ and a be as described in the/irst sentence o/Theorem 8.1 and in addition let ~ be not only metrically sta~lard but actually standard. Let there exist a Borel set S in ~(awhich meets each orbit o / ~ under ~ exactly once. Then ~ is a regularly imbedded

in ~).

U N I T A R Y R E P R E S E N T A T I O N S O F G R O U P E X T E N S I O N S . I 303 Proo/. A p p l y Theorems 5.2 a n d 7.3.

I t would be interesting t o k n o w w h e t h e r or n o t it is possible t o s t r e n g t h e n T h e o r e m 9.2 along the lines suggested b y T h e o r e m 8.6 of [13].

THEOREM 9.3. Let (~, ~ and a be as described in the/irst sentence o/ Theorem 8.1 and let ~ be a regularly imbedded in ~ . Suppose that/or each L E ~ ~ the subgroup ~L consisting of all x with L x ~_ L is such that ( ~ L / ~ ) ~ is o/ type I where eo is the essentially unique multiplier in ~L/~K de/ined by Theorem 8.2. Then ~ is o / t y p e I.

Proo/. A p p l y Theorems 8.4 a n d 9.1.

THEOREM 9.3 provides an inductive m e c h a n i s m for establishing t h e t y p e - I - n e s s of complicated groups (cf. Dixmier in [4] a n d [5]). Less directly T h e o r e m 8.4 provides such a m e c h a n i s m for establishing smoothness a n d metric smoothness. W e are n o t p r e p a r e d to f o r m u l a t e a precise t h e o r e m b u t c o n t e n t ourselves with t h e r e m a r k t h a t , generally speaking, when one has a n " e x p l i c i t " e n u m e r a t i o n of t h e irreducible a representations of a g r o u p one can m a k e use of it to show t h a t the g r o u p has a s m o o t h a dual.

F o r t h e special case in which ~ is c o m m u t a t i v e , a--- 1 a n d ~ is a semi direct p r o d u c t of with ( ~ / ~ applications of Theorems 8.4, 9.1, 9.2, a n d 9.3 t o concrete g r o u p s h a v e been described in [10] a n d [11]. I n this section we shall discuss some examples to which t h e results of [10] a n d [11] do n o t apply. We begin with a few e l e m e n t a r y facts a b o u t t h e exist- ence of n o n trivial multipliers.

T H E O R E M 9.4. Let :K and ~ be closed subgroups of the separable locally compact group (~ such that ~ is normal, :~ N ~ = e and : ~ = (~. Then every multiplier ~,' /or (~ is similar to a multiplier ~ /or ~ which may be uniquely represented in the /orm:

(a) v (x lyl, x~y2) = a (x 1, Yl (x~))~o (Yl, Y~) g (xz, Yl) where x 1 and x 2 are in ~ , Yl and y~ are in ~ , Yl (x2) = Ylx2Yl 1, (r is a multiplier/or ~ , o~ is a multiplier/or ~ , g is a Borel /unction from ~ • 74 to the complex numbers of unit modulus, g is one on 7K • e and a and g satis/y the two/ollowing identities:

(b) a (y (Xl), y (x~)) = a (xl, x~)g (x 1 x~, y))/(g (Xl, y)g (xz, y)), (e) g (x, yl Y~) = g (Y~ (x), Yi) g(x, Y2)-

Moreover/or every choice o/ a, ~o and g satis/ying (b) a n d (c) the/unction ~ defined by (a) is a multiplier/or (~.

Proo/. L e t v' be a multiplier for ~ a n d let V' be a n y v' representation. ThenV~ ~ = v ' ( x , y ) V'~ Vu for all x in :~ a n d all y in ~ . L e t V ~ = ( 1 / u ( x , y ) ) V ~ . T h e n F is a r e p r e s e n t a t i o n for a multiplier v which is similar t o v'. Since v' (x, e) = u' (e, y) = 1 for all x, y ~ :~ • ~4 it follows t h a t V~ ~ = V~ V'~ = V~ V~ = A~ B~ where A a n d B d e n o t e t h e restrie-

304 G E O R G E W . M A C K E Y

tions of V t o :K a n d ~H respectively. Moreover A is a a r e p r e s e n t a t i o n of ~ a n d B is a n (o r e p r e s e n t a t i o n of :H where a a n d to d e n o t e the restrictions of v to ~ a n d :H respectively. N o w

T h u s v (xly ~, x~y,) Az, By, A~, By, = a (x 1, Yl (Xu) )~ (Yl, Y*) A,, Ay,(,,) By, By,

a n d hence Ay,(**)ByllA,,By, is identically v(xlyi,x2y~)/(a(xl,Yl(X2))w(yl,y2) ) times t h e identity. H e n c e this last expression d e p e n d s only on x 2 a n d Yl. D e n o t i n g it b y g(x,,yl) we o b t a i n (a) a n d g (x2, e ) ~ 1. N o w let a a n d eo be a r b i t r a r y multipliers for :~ a n d 74 respectively a n d let g be a Borel function f r o m ~( • ~ to the c o m p l e x n u m b e r s of m o d u l u s one such t h a t g (x, e ) ~ 1. Define v b y (a). A s t r a i g h t f o r w a r d calculation shows t h a t v is a multiplier for (~ if a n d only if g a n d a satisfy the following identity:

(d) g (x3, Yl Y2) g (x2, Yl) a (x I Yl (x,), Yl Y2 (x3)) a (x 1, Yl (X2))

= g (x~ Y2 (x3), Yl) g (xa, Y2) a (x 1, Yl (x2) Yl Y2 (x3)) a (x 2, Y2 (xa)).

Setting y~ = e a n d using t h e fact t h a t a is a multiplier (d) reduces to (b). M o r e o v e r using (b) to simplify (d) we get (c). T h u s (d) is e q u i v a l e n t to (b) a n d (c) t o g e t h e r a n d t h e proof is complete.

W e n o t e t h a t w h e n a ~ 1 (b) reduces to the s t a t e m e n t t h a t for all y in ~ , x->g (x, y)is a h o m o m o r p h i s m of :K into t h e complex n u m b e r s of m o d u l u s one; t h a t is a m e m b e r of the g r o u p :~ of all one dimensional u n i t a r y r e p r e s e n t a t i o n s of :K. T h u s we h a v e t h e

COROLLARY. The multipliers r / o r (~ which reduce to one on ~)( and on ~ are ~ust the /unctions on (~ • ~ such that u (x 1Yl, x2y2) = ~yl (X2) where y--> Zy is a / u n c t i o n / r o m ~ to ~ such that Xy(x) is a Borel /unction o/ x and y and Zy,y, = [Zy,]Y2Xy,/or all Yl and Y2 in ~4. (Here IX/Y (x) = X (Y (x)).)

T ~ E O a E ~ 9.5. Let Y-->X.y be as described in the preceding corollary. Then the correspond-

0 / 0

ing multiplier is trivial i / a n d only i/ there exists a member Z ~ o / ~ such that Zy [X ]Y/ Z /or all y e :H.

Proo/. L e t Z ~ be a n y m e m b e r of ~ . L e t g(xy) = Z~ for all x , y in :~ • :H. L e t v be the t r i v i a l multiplier on q6 defined b y g. W e v e r i f y a t once t h a t

V (XlYl, xeY2) = g (x~ylx2y~)/g (x~y~) g (x2y,) = go (Yl (Xu))/~ 0 (X2)"

Conversely let Y-->Z~ define a trivial multiplier. T h e n t h e r e exists a Borel function g f r o m (~ to the c o m p l e x n u m b e r s of m o d u l u s one such t h a t

g (xly~ (x2)yiy~) = g (x~y~)g (x2y,)z~ ' (x~)

for all x~ a n d x~ in ~ a n d all y~ a n d Y2 in ~ . I f we set Yl = e this becomes g ( x l x , y,) =

U N I T A R Y R E P R E S E N T A T I O N S OF G R O U P E X T E N S I O N S . I 3 0 5

g(xl)g(x2y~). Thus g restricted to ~( defines a m e m b e r Z ~ of ~( a n d ' o u r original identity becomes

Z ~ (xl)Z ~ (Yl (x~)) g (Yl Y~) = Z ~ (Xi)~ 0 (X2) ~ (Yl) g (Y~)X~, (x~).

B u t since v reduces to one on ~ it follows t h a t g(YlY2)= g(Yl)g(Y2) a n d we obtain )C~, (x~) =

)~o

(Yl (x~) ) / g ~ (x~)

as was to be proved.

We shall be chiefly interested in the case in which y (x) ~ x; t h a t is in the case in which (~ is the ordinary direct product of ~ and ~ . Our results yield

THEOREM 9.6. Let ~ be the direct product o/ the separable locally ~ m p a c t groups ~)~

and ~ . Then (up to similarity) the multipliers v / o r (~ are just the/unctions on (~ • (~ such that v (xlyl, x2y~)= a (xl, x~)w (Yl, Ye)Zy, (x~) where (~ is an arbitrary multiplier/or ~(, m is an arbitrary multiplier/or ~ and y-->Zy is a homomorphism o~ ~ into ~]( such that Zy (x) is a Borel /unction on ~ x ~ . The multiplier de/ined by a, oJ and Y-~Z~ is trivial i/ and only i / a and o~ are trivial and Z y ( x ) ~ 1.

Example 1. Let ~ = ~ x ~ where ~( is a separable locally compact abelian group and is its dual. L e t v(xl,yl;x~,y~) = yl(x~) for all x 1 and x~ in ~( and all Yl and y~ in ~(. B y Theorem 9.6 v is a non trivial multiplier for the abelian group (~. L e t us determine the v representations of (~ b y applying Theorem 8.4. Since v reduces to one on J(, ~ coincides with ~ and is certainly smooth and of t y p e I. L e t ~'L denote the irreducible v representation of ~( defined b y Y0 E ~ . Since the inner automorphisms of (~ are all trivial (~~ x'~' is simply ~'L~ m u l t i p l i e d b y r ( x i - ~ , y i i ; x ~ , y ~ ) / v ( x ~ x y ~ ; x ~ , y ~ ) v ( x ~ , y ~ ; x , e ) y ~ ( x ~ ) / y s ( x i ~ ) 9 yl(x) = y ~ ( x ) ; t h a t is ~'~'-~L~. Thus there is just one o r b i t - - t h e whole of ~ . Moreover the subgroup leaving a n y one of its elements fixed is just ~(. Since ~(/~( is the identity there is just one irreducible v representation of (~ associated with this orbit. I n other words (~

has to within equivalence just one irreducible v representation. I t is infinite dimensional and is the v representation induced b y the identity representation of ~(.

I t is interesting to compare example 1 with Theorem 1 of [9] which presents exactly the same result from a rather different point of view.

E x a m p l e 1 points up three ways in which the theory of projective representations differs sharply from the theory of ordinary representations. An abelian group can have infinite dimensional irreducible projective representations and for a given multiplier v can have a unique irreducible v representation. The v representations of a direct product need not be related in a simple w a y to the v representations of the factors even when the factors have t y p e I ~ duals. Of course if v is a direct product of multipliers for the factor groups; t h a t is if the Y-~Z~ of TheOrem 9.6 is identically one then it is not difficult to p r o v e an analogue of the corollary to Theorem 1.8 of [12].

3 0 6 GEORGE W. MACKEu

Example 2. L e t ~ be as in e x a m p l e 1 b u t let ~ be r e p l a c e d b y ~ w h e r e ~ is a c o u n t a b l e , d i s c r e t e l y t o p o l o g i z e d , dense s u b g r o u p of ~ . L e t v n o w d e n o t e t h e r e s t r i c t i o n to ~ • of t h e r of e x a m p l e 1. J u s t as before we h a v e (~~ ''y' = ~*~'-'L x b u t n o w yl can no l o n g e r be a n a r b i t r a r y e l e m e n t of ~ b u t is r e s t r i c t e d to lie in :~. T h u s t h e r e a r e m a n y o r b i t s - - o n e for e a c h ~/ coset in ~ . J u s t as before t h e r e is e x a c t l y one i r r e d u c i b l e v r e p r e s e n t a t i o n of for e a c h of t h e s e orbits; n a m e l y , t h e one i n d u c e d b y a n y m e m b e r of t h e o r b i t . H o w e v e r in t h i s case t h e i r r e d u c i b l e v r e p r e s e n t a t i o n s d e s c r i b e d in T h e o r e m 8.4 do not e x h a u s t t h e i r r e d u c i b l e v r e p r e s e n t a t i o n s of ~ . I n a d d i t i o n t o t h e o r b i t s in :~ t h e r e is a t l e a s t one p r o p e r quasi o r b i t . H a a r m e a s u r e in :~ is ergodic u n d e r t h e g r o u p of t r a n s l a t i o n s b y m e m b e r s of ~ / a n d hence defines a n ergodic i n v a r i a n t m e a s u r e class n o t c o n c e n t r a t e d in a n y o r b i t . T h e e x i s t e n c e of t h i s quasi o r b i t c a n be u s e d t o show t h a t ~ h a s p r i m a r y v r e p r e s e n t a t i o n s w h i c h a r e n o t of t y p e I in a d d i t i o n t o m a n y i r r e d u c i b l e v r e p r e s e n t a t i o n s o t h e r t h a n t h o s e d e s c r i b e d in T h e o r e m 8.4. T h u s a c o m m u t a t i v e ~ can h a v e n o n t y p e I p r o j e c t i v e r e p r e s e n t - ations. W e shall defer d e t a i l s t o o u r p r o j e c t e d p a p e r on t h e i n t r a n s i t i v e case.

W h e n :~ is a v e c t o r g r o u p , T h e o r e m 1 of [9] r e d u c e s t o t h e t h e o r e m of S t o n e a n d y o n N e u m a n n on t h e u n i q u e n e s s of sets of o p e r a t o r s s a t i s f y i n g t h e H e i s e n b e r g c o m m u t a t i o n relations. H e n c e E x a m p l e 1 a b o v e c o n t a i n s t h i s t h e o r e m . W e shall n o w i n d i c a t e a s i m i l a r c o n n e c t i o n b e t w e e n E x a m p l e 2 a n d t h e p r o b l e m of f i n d i n g all sets of o p e r a t o r s s a t i s f y i n g t h e " a n t i c o m m u t a t i o n r e l a t i o n s " of q u a n t u m field t h e o r y . Tl~e p r o b l e m is t h a t of f i n d i n g all sequences A1, A2, ... of b o u n d e d o p e r a t o r s on a H i l b e r t space ~ such t h a t A t A k + A k A l = 0 a n d A j A ~ + A ~ A t =6t~ for all ?', k = 1, 2 . . . F o l l o w i n g H . W e y l ([16], p a g e 2 5 2 ) we l e t A i = 89 (P2r + iP21) w h e r e i 2 = - 1 a n d t h e P i a r e self a d j o i n t . W e n o t e t h a t t h e a n t i c o m m u t a t i o n r e l a t i o n s e x p r e s s e d in t e r m s of t h e P j ' s a r e P l 2 = 1 a n d P j P k + P ~ P j = 0 for all ]:~ k. F o r o p e r a t o r s T w i t h T 2 = 1, u n i t a r i n e s s is e q u i v a l e n t t o self a d j o i n t n e s s . T h u s o u r p r o b l e m is t h a t of f i n d i n g all sequences P1, P~ . . . . of u n i t a r y o p e r a t o r s such t h a t P~ = 1 a n d P i P e + PkP~ = 0 for ~ ~: k. G i v e n such a sequence l e t S 1 = P1 a n d l e t S j = i P t - l P i for j = 2, 3 . . . T h e n t h e S t f o r m a sequence of u n i t a r y o p e r a t o r s such t h a t S j S t + 1 = - S~+IS j a n d S i S j + k = Sj+eS~ for all ~ = l , 2 . . . . a n d all k = 2, 3 . . . M o r e o v e r P i = ( - i) j-1- (S 1S~ ... Sj) for ~ = 2, 3 . . . C o n v e r s e l y ff S1, S~ . . . . is a sequence of u n i t a r y o p e r a t o r s s a t i s f y i n g t h e c o n d i t i o n s j u s t e n u n c i a t e d a n d we let P s = ( - i f - 1 ( S I S ~ . . . Sj) for ] = 2, 3 . . . . a n d P1 = S~ t h e n P~ = 1 a n d P i P e + P k P i = 0 for j ~: k. I n o t h e r w o r d s o u r p r o b l e m is also e q u i v a l e n t t o t h a t of f i n d i n g all sequences S 1, $2, ... of u n i t a r y o p e r a t o r s such t h a t S~ = 1, S j + I S j = - SiSi+ ~ a n d S~Si+ ~ = S~+~S~ for all ~ = l , 2 . . . . a n d k = 2, 3, .... L e t ~ b e t h e d i r e c t p r o d u c t of c o u n t a b l y m a n y g r o u p s of o r d e r t w o a n d l e t x2~+~ d e n o t e t h e g e n e r a t o r of t h e j t h g r o u p . F o r e a c h j = 1, 2 . . . . l e t y ~ d e n o t e t h e u n i q u e e l e m e n t of :~ such t h a t y2~(x~+~) = 1 w h e n e v e r 12i - 2k + 1] * 1 a n d ya,(x~,+~) = - 1 w h e n e v e r 12~ - 2k + 11 = 1.

U N I T A R Y R E P R E S E N T A T I O N S OF G R O U P E X T E N S I O N S . I 307 L e t 74 denote the subgroup of :K generated b y the y~j. L e t v be the multiplier on :~ • described above. A v representation L of ~ x ~ is uniquely determined b y its values on the xaj+l and the YaJ. Let $1, $2, ... be a sequence of u n i t a r y operators. I t is easy to see t h a t there exists a v representation L of :~ • ~ such t h a t Lxai+ 1 = Saj+l and L~2 j = Sat for all

= 1, 2 . . . . if a n d only f f t h e Sj form a solution of the second reformulation of our problem.

Thus the problem of finding all sets of operators satisfying the "anti c o m m u t a t i o n relations"

is equivalent to the problem of finding all v representations of the discrete group ~( x ~ . Since this representation problem is a special case of t h a t considered in example 2 it involves the consideration of non transitive quasi orbits and hence is only p a r t l y solved b y the theory of the preceding sections. A partial solution going beyond t h a t provided b y our theory has been sketched in a recent note of Ghrding and W i g h t m a n [6]. We hope t h a t our projected investigation of the non transitive case will yield general results whose application to the case at hand will include the results of [6]. We r e m a r k t h a t example 2 was suggested to us b y our study of [6].

Example 3. L e t (~ be the group of all 3 x 3 unimodular real matrices which are zero above the main diagonal. Let :~ be the normal subgroup of (~ consisting of all m e m b e r s which are one on the main diagonal. L e t ~ be the subgroup of (~ consisting of all members which are zero off the main diagonal. We shall determine the (ordinary) irreducible representations of ~ and then those of (~. (~ is clearly a semi direct product of :~ and ~ . L e t (a, b, c) denote the m a t r i x 1 and let (2,/~, v) denote the m a t r i x # 0 . Then

e 0 v

every m e m b e r of (~ is uniquely of the form (a, b, c~ (2,/u, v) where a, b, c, 2,/~, and are real numbers such t h a t 2#v = 1. We compute t h a t

(al, bl,Cl) (aa, ba, ca) = (a 1 + a 2 + clb a, bl + b a, el + ca~, t h a t (21,Jul ~1) (22,/f~2,~'2) = (~l/~2,/A1/Aa,~:lY2)

and t h a t (2,/~,v)(a,b,c) (2,/u,v)-l=(~a,~b,V-c~/ 9

\/~ A /A /

To determine the irreducible representations of ~ we note t h a t the center E of :~ is the set of all ( a , 0 , 0 ) a n d take this normal subgroup of :~ as the :K of Theorem 8.4. The quotient group ~ / ~ acts trivially on E and hence on 5. Thus the orbits in ~ are the points of and the groups holding the points of ~ fixed all coincide with (~. Now the points of ~ are in one-to-one correspondence with the real numbers in such a fashion t h a t the real n u m b e r r corresponds to the representation ( a , 0 , 0 ) - - > e x p (Jar). L e t ]~(a,b,c)=exp(ir(a-bc)).

We compute t h a t /, ((al, bl, cl) (a~, ba, ca)) = exp ( - irb 1 ca) h ((al, b~, c~))/, ((a a, b a, c~) ). Thus

3 0 8 G E O R G E Vr M A C K E Y

[r defines a one dimensional projective representation of Y~, with multiplier exp(-irblc2) , which reduces on ~ to the representation defined by r. Hence the most general irreducible representation of 3( is obtained by choosing an r and an irreducible projective representation V of : ~ / Z whose multiplier is exp (irblc2) and then forming ]r V. When r = 0 we get simply the one dimensional representations of :K defined by the one dimensional representations of the two dimensional vector group :K/E; t h a t is the representations W s'* where W~'~,b,c>

is multiplication by exp(isb + itc). When r ~= 0 the multiplier exp (irblc2) of ~ / ' Z is of the form discussed under Example 1. Thus there is a unique irreducible projective representation V r of ~ / Z with the multiplier in question and hence a unique irreducible representation W r = ]r V r of :7( associated with the orbit of (a, 0, 0)--> exp (far). I n all then, we have one two parameter family s,t---> W s't of one dimensional representations of ~ and one discon- nected one parameter family r--> W ~ (r :~ 0) of infinite dimensional irreducible representations of ~ . I t follows easily t h a t :K has a smooth type I dual.

To determine the irreducible representations of (~ we shall first determine the orbits in ~ under the action of ~ . W ~'t ~,~,~)<~,b,c,>~,.~,~ , is multiplication by exp i ( ( S ; b + ~ - / ! t v c i t and hence is equal to ,, <~,b,o> 9 Thus the one dimensional representations fall into four orbits as follows. 0~ contains all W ~'t with s ~= 0 and t ~= 0. 02 contains all W ~'~ with s =~ 0, 03 contains all W ~ with t=~0, 04 contains W ~176 only. Now W~,,.~><~,0,0>(~,,,~)-,= W~a/~,o,0>

which is multiplication by exp (i (r/2) ar). Hence W ~ (~,~,~)<a,b,~,>(~.,.,~)-, = ,, <~ ~ ~> t h u s all in- uz~ r/z finite dimensional irreducible representations of ~ lie in a single orbit 0~. The subgroup of ~ leaving W L1 invariant is the set of all (2,/~,v) with ~ = / t = v and ~3 = 1; that is it con- sists of the identity alone. There is then a single irreducible representation associated with 01. I t is infinite dimensional and is the representation of (~ induced by a n y one dimensional representation W s't of :K with s ~ 0 and t ~- 0. The subgroup O2 of ]D leaving W 1'~ invariant is the set of all (z,/~,0 with/~ = 2 and 2 / ~ = 1; that is, the set of all (2,2,1/22) and is isomorphic to the multiplicative group of all non zero real numbers. The possible extensions of W ~' t to ~ 2 are in an obvious one-to-one correspondence with the one dimensional representations of ~ and hence with the non zero real numbers. The irreducible representations of (~ associated with 0 , are the irreducible representations induced by these extensions. They form a family of infinite dimensional representations parameterized by the non zero real numbers. The ::'reducible representations of (~ associated with 03 are constructed analogously; the only difference being that ~ is replaced by the group ~3 of all (1/22,2,2).

_Again we get a family of infinite dimensional representations parameterized by the non zero real numbers. The irreducible representations of (~ associated with 0~ are in an obvious one- to-one correspondence with the irreducible representations of the abelian group ~ . They are

U N I T A R Y R E P R E S E N T A T I O N S OF G R O U P E X T E N S I O N S . I 309 all one dimensional and m a y be parameterized by the pairs of non zero real numbers. The subgroup ~ r of ~ leaving W 1 invariant is the set of all (2, 1/22, 2). I t is not hard to show that this group has no non trivial multipliers and hence t h a t W 1 m a y be extended to be a representation of :KO~r The possible extensions correspond to the irreducible representations of

~ . As with 02 and 03 we get a family of infinite dimensional irreducible representations parameterized by the non zero real numbers. This time however the inducing representations are themselves infinite dimensional. I n all we have one isolated infinite dimensional representation, three families of infinite dimensional representations, each parameterized by the non zero real numbers and one family of one dimensional representations parameterized by the pairs of non zero real numbers.

E x a m p l e 4. Let 74 be a finite group and let • be group of automorphisms of ~4. Let be the group of all functions / from the integers to 74. Equipped with the direct product topology :~ becomes a compact group. For each integer n o and each a E.~ let ~, n o denote the a u t o m o r p h i s m / - - > / ' w h e r e / ' (n) = a (/(n + no) ). The set of all of these automorphisms is a group a isomorphic to the direct product of A with the additive group of all integers.

Let ~ denote t h a t semi direct product of the compact group :K with the discrete group a in which the homomorphism from a into the group of automorphisms of :K is the natural one. If n 1 and n 2 are integers with n I < n 2 then the set of all / in J( with / (n) = e for n 1 ~ n ~< n~

is a normal subgroup whose quotient is naturally isomorphic to the direct product of a finite number of replicas of 74. The representations of this quotient group define representations of ~ and it follows easily from the theory of compact groups that every irreducible representation of :~ m a y be so obtained (with varying n 1 and n 2 of course). Thus the irreducible representations of :~ m a y be put in a natural one-to-one correspondence with those functions M , n - - > M n from the integers to the irreducible representations of :H such t h a t M n is the one dimensional identity for all but finitely m a n y values of n. I t is clear that the orbits of ~ under a are all infinite except for the one containing the one dimensional identity representation. Thus all irreducible representations of (~, except those trivially derived from representations of (~/:K - a , are infinite dimensional. The subgroup of a taking the representation defined by n---~M '~ into one equivalent to itself is the set of all ~, 0 such t h a t a takes M n into a representation equivalent to itself for all n. Hence to determine the irreducible representations of (~ it suffices to determine the irreducible representations of the finite group :H, study the way in which the automorphisms in A act on subsets of these representations and determine the a representations of certain subgroups of the finite group ~4 for certain values of a. If we take ~ to be A6, the alternating group on six elements, and ~4 to be the group of all automorphisms of ~4 it is easy to see that all of these things m a y be done quite explicitly and t h a t non trivial a's arise. We shall content ourselves

310 GEORGE W. MACKEY

here with an indication of the proof t h a t non trivial a ' s arise. Let $6 denote the symmetric group on six elements so t h a t As is a normal subgroup of index two of ts. As shown on page 209 of [2] the automorphisms of Ms induced by the inner automorphisms of t s form a normal subgroup Ms of index two in the group M of all automorphisms. Now it follows from the character table on page 266 of [8] t h a t amongst the irreducible representations of $6 there are (to within equivalence) just two of dimension ten and it is easy to see t h a t these remain irreducible and become equivalent when restricted to As. L e t W denote the representation of :K defined b y the function n - - + M '~ where M ~ is the ten dimensional irreducible representation just described and for j ~: 0, M j is the one dimensional identity representation. The subgroup of a taking W into a representation equivalent to itself is the group of all ~, 0 and ~ is restricted to the subgroup of M taking M ~ into something equivalent to itself. Moreover inspection of the character table of M s (easily derived from t h a t of Se) shows t h a t this later subgroup is the whole of M. The representations of (~ associated with the orbit of W are thus in a natural one-to-one correspondence with the a representations of M for some a. I f this a were trivial there would exist an ordinary representation of J(M extending W. Hence there would exist an ordinary representation of MsM extending M ~ Let L be a n y such representation of MsM. B y the character table for t s and the defini- tion of M ~ we know t h a t L has exactly two inequivalent extensions to the normal subgroup of M defined b y the inner a u t o m o r p h i s m of ts. Moreover from the proof given in [2] t h a t there exist automorphisms of Ms other t h a n those in Ms it is easy to see t h a t these automorphisms interchange the two extensions of L to Ms. I t follows a t once from the general theory t h a t there can be no extension of L to M and hence no extension of W to ~ M . Thus a cannot be trivial. Thus when ~ is non c o m m u t a t i v e non trivial multipliers can occur even if (~ is a semi direct product of ~ and (~//~.

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