Let G be a Lie group, G " its unitary dual space consisting of. the set of all equivalence classes of continuous irreducible

Transcription

1 INTRODUCTION Let G be a Lie group, G " its unitary dual space consisting of the set of all equialence classes of continuous irreducible unitary representations of G endowed with the hull-kernel (Fell) topology. The orbit space 1 * /G, under the coadjoint action of G '- * on the real dual space a of the Lie algebra ~ of G, is gien the quotinent topology fro )*. If G is an exponential Lie group, i.e. the roots of the adjoint representation of ~ are of the for (1+it)f, ter, fe * ~ the Kirillo correspondence is a bijection of ':! * /G onto G, " and Pukanszky has shown that the ap C) * /G+G" is continuous, [11 ~ Proposition 1 ]. It is still an open " * question if the inerse ap O:G+ ~ /G is continuous. For nilpotent G this was settled in [4], and for *-regular exponential groups in [3 ]. See also [6]. In the present article we show that the restriction of 0 to the subspace G. of G, consisting of all equialence classes of infinite diensional repr~sentations,' is in fact continuous. NOTATIONS. Throughout the paper we shall use basic results fro the theory of induced representations of Mackey, [9], which we assue known to the reader. If K is a closed subgroup of the Lie group G, and S a unitary representation of K, ind~(s) denotes the unitary representation of G induced fro S.

2 - 2 - CONTINUITY ON G '"' GO. THEOREM. G.* Let G be an exponential Lie group, J the dual of its Lie algebra. Then the restriction of the Kirillo ap to the subspace <~*/G)GO of the coadjoint orbit space, consisting of the set of all orbits that correspond to eleents of G '"',. GO l.s bi- continuous, where these subspaces are assued to be equipped with the relatiized quotient topology and hull-kernel topology, respectiely. PROOF. Continuity of the ap G..* '"' d /G+G was shown in [11~ Proposition 2]. Let '"' c..* Q;G+ d /G '"' QIG GO denote the inerse ap. We proceed to show is continuous, assuing inductiely that the result is true for all exponential Lie groups of diension saller than Let denote a sequence of eleents in G '"' di(g). that conerges to T 0, T 0 EG..,. '"' We are going to proe that a subsequence of {Q(T)}:=l conerges to the orbit Q(T 0 ) in the topology of G..*/G ( c.."*/g) ; relatiized to J We fix a closed noral connected 00 subgroup N of codiension one in G. Applying the Mackey achine to N and G, [9~ Theore 8.1 ], we shall partition the proof according to the. following four cases 1 taking a subsequence of (I) The restriction s =T IN is irreducible for each =O, 1, 2 1 In this case T is an extension of S,for eery. By continuity of the restriction a.p we hae S +S in N 0 '"' oo and by our inductie hypothesis the sequence of orbits {Q(S)}:=l "*. conererges to O(s 0 ) in (1'\: /N). Hence there exist functionals "'

3 - 3 - f in "(S ) I =O ,..., such that f.. f. Let V= Re be a 0 copleentary subspace to )1. in ~.J =n+ i and let e ']II in ')"* be the eleent dual to e. We shall identify 'Yt'* with a subspace ~.,. ~'*= * '* of by eans of the relation Re +'n... Thus, regarding f as an eleent of 1 '*, let s - be the class in G " corresponding - to the orbit of f I =O I 1 I 2 I. s is known to be an extension of S hence, by Mackey, there exists a character X of G, with T =s - x. Now conerges to * in ~, and therefore S ~ s 0 by continuity of the Kirillo ap, [11; Proposition 1 ]. We fix an infinite diensional Hilbert space H with a countable basis. The space of all irreducible unitary representations of G on H, denoted by "" as in [5;18.1.9], then the canonical ap Irr (G)~G" Irr (G), is topologized which assigns to each representation its unitary equialence class, is continuous and open, [5:3.5.8]. Therefore we can find representations - - t,s in Irr (G) of classes T,S respectiely, so that t ~ t 0 and 5~5 0. Tbiseans and - - I s (g) - s 0 (g) I ~ 0 1 uniforly on copacta in G, for all in H, where we let I R denote the nor in H, [5; Proposition ]. Let EH, ge G. Then lx 0 {g)-xin(g) l ll = x 0 (g)s 0 (g)-x(g)s 0 (g)l < lx 0 (g)s 0 (g)-x(g)s(g)l + x(g)s(g)-x(g)s 0 {g)n = x0 (g)s0 (g)-x(g)s(g)l + ls(g)-s0 (g)l ~ 0,

4 - 4 - uniforly on copacta in G. It follows that the sequence of characters conerges to x 0. Let h be the functional '-* in d corresponding to X : rn=o I 1 I 2 I By the aboe f +h lies in the orbit C{T ) and f +h + f 0 +h 0. Therefore we hae shown that O(T )+C(T 0 ). (II) Each T 1 = , is induced fro a representation 5 1. dg {. I) of N 1 T =1n N S. By the Mackey achine the restrictions T I are supported on N the orbits G S " 1 in N under the action of G by conjugation. Now TlN ~ TOIN1 hence we can find a sequence {x}~=o of eleents in G such that T =indng { S ) 1 =O 1 1, 2 1 x S 1 + x S 1 We put 0 0 S =x S 1 : then (the stability groups under G are all equal to N). Applying our inductie hypothesis we obtain Q(S )+C{s 0 ) in r /N: thus we can find" functionals rfl * with f + f 0. Using the fact that that O(T )+O(T 0 )...J. f +n cq(t ) f in Q(S ) we conclude {III) T 0 jn=s 0 is irreducible and each induced fro a representation S 1 of T 1 = is N 1 T = i nd GN ( S 1 ) Arguing as in {II) we can find representations s in the orbit of TiN 1 =l~2,3,... 1 such that T = i ndng ( S ) 1 and by our inductie hypothesis, functionals f in Q( s ) =0,112, 1 with f ~ f 0. As in (I) T 0 =s 0 x 0 where s 0 denotes the eleent in G~ " corresponding to f 0 : and the character is identically one on N1 and is gi~n by a functional Using that T is induced fro S, >O, we hae hence Q ( T ) +Q (TO ). ho J. c < T >~f +n, f+h 0 EO{T), and clearly f+h 0 ;. f 0 +h 0. We hae shown that

5 - 5 - (IV) and T I =S is irreducible, =1,2,3,... N the description of the hull-kernel topology it follows that the sequence {S } =l conerges to each eleent in the G-orbit of S + s 0 : and by irtue of the inductie hy Fro pothesis we can find functionals f in Q ( S ) t =Q t With f + f 0. For each =0,1,2,..., Irr (G) in the class of T, so that Irr (G) " (the canonical ap we let t { t} :=1 is a Cauchy sequence in I rr (G) we see as in case {I) that for t t 0 be an eleent of in the topology of Irr (G)+G is open). In particular <X> EH, geg, and, writing t =s x, lx (g)-x (g)l ll 'lx (g)s (g)-x (g}s (g)l +Is (g)-s (g)l n n n n Using this together with the fact that {~\:= 1 is a Cauchy sequence in Irr (G) (recall that s corresponds to the functional f of 'J ), we hae { x } is a Cauchy sequence of =l characters of G. Let x 0 be its liit character and denote by h the functionar of ~ ;>O. Thus h + h 0, and therefore f + h f 0 + h 0 We hae proed Q(T)+Q(T 0 ). QED For nilpotent Lie groups bicontinuity of the Kirillo ap was first proed by Brown, [4]. This result is also a consequence of the aboe,theore. COROLLARY. Let G be a siply connected and connected nilpotent Lie group. Then the Kirillo ap is a hoeoorphis of the coadjoint. orbit space ~~/G d onto the unitary dual space G. "

6 - 6 - PROOF. The ap is clearly a hoeoorphis when restricted to the '- */G closed subspace of J consisting of those orbits which correspend to characters of G. Cobining this with the aboe theore, all that reains to be proed is continuity of the inerse ap G ~) */G at the identity representation, and this was shown in [10, Theore l ]. We gie a short arguent. Let >0 be gien. For 6>0 and K a closed subgroup of G, we denote by V(O,K) the neighbourhood of 1 in K consisting of all s such that for C c K, C copact, there exists a ector in the Hilbert space H(S) of S with j<s(x),>-ll<o, for all x in c. Now let TEG, and suppose TEV(o,G). Then since G is nilpotent, there exists a closed noral connected subgroup N of codiension one in G such that T is induced fro an irreducible S of N. By Mackey, T=ind~(xS) for eery x in G. And by continuity of the restriction ap and the fact that TIN is supported on the G-orbit G S, we hae x SEV(6.N) for soe x in G. Assuing inductiely /\ *; that N ~n N is continuous at the identity representation, we ay conclude by proper choice of o that the distance fro the orbit Q ( x S) to 0 in ~:_ * is less than. Now this distance is greater than the distance fro Q(T) to 0 in 'J*, and it follows that c;."' G ~I /G is continuous at 1, copleting our proof. QED

7 - 7 - AN EXAMPLE Let ~=~, 9 (0) denote the exponential Lie algebra gien by the nonzero basis relations [e 1,e 2 ]=e 2, [e 1,e 3 ]=-e 3, [e 2,e 3 ]=e 4, and let { e:} 1= 1 be a basis for 'd * dual to { ei} 1= 1 The nilradical N=exp(Re 2 +Re 3 +Re 4 ) of the group G=exp~ is isoorphic with the three diensional Heisenberg group. For *O, let S be the eleent of ~ associated to the functional of n. The isotropy group of s is all of G, and s extends to G. We s s to e 4 denote by the extension of that corresponds * in 'J*. All the extensions of s are of the for T =S "Xa where a, x a is the character.of G, equa 1 to 1 on N, with function a 1 * ae1, aer. The orbit Q a, of T a, is seen to be the hyperboloid with a distance 12ii'V fro zero, wheneer O<<a. Hence a sequence [Q t a n' n and a ++co 1 n of such orbits ay conerge to 0 een if let e. g. =1 /n 2 n representations, f T ) l a: '. n n. ' and an=n. Then the two sequences of {s }, both conerge to lg by n continuity of the Kirillo corresp6ndence. Still the sequence of characters is unbounded. This answers in the negatie a question raised at the end of [6]. +0 n

8 - 8 - REFERENCES 1. Auslander, L., Kostant, B., Polarizations and unitary representations of solable Lie groups, Inentiones ath. 14 {1971), Baggett, L., Kleppner, A., Multiplier representations of abelian groups, J.Funct.Anal., 14 (1973), Boidol, J., ~-Regularity of Exponential Lie Groups, Inentiones ath. 56 (1980), Brown, I.D., Dual topology of a nilpotent Lie group, Ann.Sci. Ecole Nor. Sup., (4) 6 ( 1973), Dixier, J., Les C -algebres et leurs representations, Gauthier-Villars, Paris Fujiwara I H. I Sur le. dual d I un groupe de Lie resoluble exponentiel, J.Math.Society Japan, 36 (1984), Joy, K.I., A description of the topology on the dual space of a nilpotent Lie group, Pac.J.Math.,. 112 (1984), Kirillo, A.A., Unitary representations of nilpotent Lie groups, Usp. Math.Nauk 17 (1962), = Russian Math. Sureys, 17 (1962), Mackey, G.W., Unitary representations of group extensions, Acta Math., 99 (1958), Ol'shanskii, G.I., The topology of the space of unitary representations of a nilpotent Lie group, Functional. Anal. i Priloien., 3 (1969), no.4, =Functional Anal.Appl. 3 ( 1 969) o 11. Pukanszky, L., On unitary representations of exponential groups, J.Func.Anal., 2 (1968),