A CONTINUITY PROPERTY OF THE COADJOINT ORBIT CORRESPONDENCE OF EXPONENTIAL LIE GROUPS

the set of all equivalence 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 given the quotinent topology from

)*.

If G is an exponential Lie

group, i.e. the roots of the adjoint representation of ~ are of the form (1+it)f, tER, fE

*

~ the Kirillov correspondence is a bijection of

':! * /G

onto G, and Pukanszky has shown that the map

"

* "

C)

/G+G is continuous, [11 ~ Proposition 1 ] . It is still an open

" *

question if the inverse map ^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. ^1\ of G, ^1\ consisting of all equivalence classes of infinite dimensional repr~sentations,' is in fact continuous.

NOTATIONS. Throughout the paper we shall use basic results from the theory of induced representations of Mackey, [9], which we assume 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 from S.

'"'

CONTINUITY ON G . GO

THEOREM. Let G be an exponential Lie group, ^G.* J the dual of its Lie algebra. Then the restriction of the Kirillov map to the subspace <~*/G)GO of the coadjoint orbit space, consisting of the set of all orbits that correspond to elements of G ,

'"'

GO l.S

.

bi- continuous, where these subspaces are assumed to be equipped with

the relativized quotient topology and hull-kernel topology, res- pectively.

PROOF. Continuity of the map ^G..* d /G+G ^'"' was shown in [11~ Proposi- tion 2]. Let _Q;G+ '"' c..* _d

/G

denote the inverse map. We proceed to show

'"'

QIG GO is continuous, assuming inductively that the result is true for all exponential Lie groups of dimension smaller than dim(G).

Let denote a sequence of elements in G

'"'

_a> that

'"'

converges to T_{0 ,} T₀EG..,. We are going to prove that a subsequence of {Q(Tm)}:=l converges to the orbit Q(T_{0 )} in the topology of

G..*/G ; relativized to ( c.."*/G) • J ₀₀ We fix a closed normal connected subgroup N of codimension one in G. Applying the Mackey machine to N and G, [9~ Theorem 8.1 ], we shall partition the proof according to the . following four cases ^{1 ·} taking a subsequence of

(I) The restriction

s

_{m m} =T

IN

is irreducible for each m=O, 1 , 21 • • •

In this case Tm is an extension of Sm ,for every m.

'"'

By continuity of the restriction ma.p we have S +S in N

m 0 ^oo

and by our inductive hypothesis the sequence of orbits {Q(Sm)}:=l conververges to O(s_{0 )} in "* .

(1'\: /N) . Hence there exist functionals

"'

f in "(S ^{) I} m=O ¹ 1 ¹ 2, ... , such that f .. f . Let V= Re be a

m m m m 0

complementary subspace to ^)1. in ~.J ⁼ⁿ⁺

vi

and let e ^']II in ')"*

be the element dual to e. We shall identify 'Yt'* with a subspace

~.,. ~'*=

* '*

of by means of the relation Re +'n.... Thus, regarding f m

1

^'*,

- _"

as an element of let

s

be the class in G corresponding

m

-

to the orbit of f I m=O ^I 1 ^I 2 ^{I • • •}

. s

is known to be an extension

m m

of S hence, by Mackey, there exists a character X of G,

m m

with _{T =s •x .}

-

_Now

m m m converges to

in ~

*

, and therefore S ~

s

₀ by continuity of the Kirillov map,

m m

[11; Proposition 1 ]. We fix an infinite dimensional Hilbert space

H _a> with a countable basis. The space of all irreducible unitary

representations of G on H , denoted by

"" Irr (G), is topologized

a>

as in [5;18.1 .9], then the canonical map ^{Irr (G)~G}

"

which assigns

a>

to each representation its unitary equivalence class, is continu- ous and open, [5:3.5.8]. Therefore we can find representations

- -

t ,s in Irr (G) of classes T ,S respectively, so that

m m ^m m m

tm ~ t ₀ and 5m~5

0

^{. Tbismeans}

and

- -

I s m (g) v- s 0 (g) vI ^~ 0 ¹ m

uniformly on compacta in G, for all v in H , where we let I •R

a>

denote the norm in H , [5; Proposition 18. 1 . 9]. Let vEH , gE G.

CD CD

Then

lx 0 {g)-xin(g) ^l•lvl = •x 0 (g)s 0 (g)v-xm(g)s 0 (g)vl

< lx 0 (g)s0 (g)v-xm(g)sm(g)vl + •xm(g)sm(g)v-xm(g)s 0 {g)vn

- - - -

=

•x 0 (g)s 0 (g)v-xm(g)sm(g)vl + lsm(g)v-s 0 (g)vl ^~ 0 ,

uniformly on compacta in G. It follows that the sequence of

characters converges to

x

_{0 . Let} ^h be the functional

m

in '-* d corresponding to X : rn=O ^I 1 ^I 2 ^{I • • •}

m By the above f +h

m m

lies in the orbit C{T ) and f +h + f₀+h_{0 .} Therefore we have

m m m m

shown that O(T )+C(T_{0 ).}

m m

(II) Each T 1 m=0 1 1 121 ••• , is induced from a representation 5 ¹

m m

. dG {. I)

of N¹ T _m =1n N S _m .

By the Mackey machine the restrictions T

I

are supported on

m N

"

the orbits G•S¹ in N under the action of G by conjugation.

m

Now TmlN

~

TOIN1 hence we can find a sequence

{xm}~=O

^of

elements in G such that T =indNG { S ) 1 m=O 1 1 , 2 1 •••

m m

x •S¹ + x •S^{1 •} We put

m m m 0 0 S =x • S ^{1 :} then m m m

(the stability groups under G are all equal to N). Applying our inductive hypothesis we obtain

rfl

*

Q(S )+C{s_{0 )} in ·rv /N: thus we can find" functionals

m m f in Q(S )

m m

with f + f_{0 .} Using the fact that .J.

m m · f

+n

^{cQ(T )} we conclude

m m

that O(T )+O(T_{0 ).}

m m.

{III) T0 ^jN=s0 is irreducible and each induced from a representation S¹ of

m

T 1 m= 1 1 2 1 3 1 ••• 1 is m

N ¹ T _m

=

^{i nd GN ( S} _m ^{1 ) •}

Arguing as in {II) we can find representations

s

m in the orbit of TmiN 1 m=l~2,3, . . . 1 such that T = i ndNG ( S ) ¹

m m

and by our inductive hypothesis, functionals f in

m ^Q(

s )

¹

- -

m

m=0,112, ••• 1 with fm ~ f 0 . ^{As in} (I) T0^=s0

·x

₀ ^{where s}₀ ^de-

"

notes the element in G~ corresponding to f_{0 :} and the character is identically one on N1 and is ^giv~n by a functional

Using that T

m is induced from S , m>O, we have m

ho·

c

^{< T} _m >~f _m

+n ,

J.

hence fm+h0EO{Tm), and clearly fm+h0 ;. ^f0^+h0 . We have shown that

Q ( T ) +Q (TO ) . m m

(IV) and T

I

=S is irreducible, m=1,2,3, ...

m N m From

the description of the hull-kernel topology it follows that the sequence {S

}m

converges to each element in the G-orbit of

m m=l

S ⁺

s

_{0 : and} by virtue of the inductive hy- m m

pothesis we can find functionals f in ^{Q (} S ) t m=Q ¹ 1 ¹ 2 1 • • • t With

m m

f _m + _m f 0 . For each m=0,1,2, ... , we let ^t

m be an element of Irr (G) in the class of

CD T , so that

m ^{t m m • t0} in the topology of Irr (G)

CD (the canonical map Irr (G)+G

"

<X> CD is open). In particular

{ tm} :=1 is a Cauchy sequence in ^I rr (G)

CD and, writing t _{m m m} =s

·x ,

we see as in case {I) that for vEH , gEG,

CD

- - - -

lx (g)-x (g)l•lvl

'lx

(g)s (g)v-x (g}s (g)vl +Is (g)v-s (g)vl

m n m m n n m n

Using this together with the fact that {~m\:=

1

is a Cauchy sequence in Irr (G)

CD (recall that s corresponds to the m

functional f of

'J ·),

^{we have { x· }·CD} is a Cauchy sequence of

m ·· m m=l

characters of G. Let x0 be its limit character and denote by h m the functionar of

~·

m;>O. Thus

f + h • f0 + _{h 0}

m m m

We have proved Q(T)+Q(T 0 ). _{m m}

h _{m m} + h 0 , and therefore

QED

For nilpotent Lie groups bicontinuity of the Kirillov map was first proved by Brown, [4]. This result is also a consequence of the above,theorem.

COROLLARY. Let G be a simply connected and connected nilpotent Lie group. Then the Kirillov map is a homeomorphism of the

.

~~/G

coadJoint orbit space ^d onto the unitary dual space G.

"

PROOF. The map is clearly a homeomorphism when restricted to the '- */G

closed subspace of J consisting of those orbits which corre- spend to characters of G. Combining this with the above theorem, all that remains to be proved is continuity of the inverse map

1\

*

G ~) /G at the identity representation, and this was shown in [10, Theorem l ]. We give a short argument. Let £>0 be given. 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 ^1\

s

such that for C c K, C compact, there exists a vector v in the Hilbert space H(S) of S with j<S(x)v,v>-ll<o, for all x in

c.

Now let TEG, and suppose 1\ TEV(o,G). Then since G is nilpotent, there

exists a closed normal connected subgroup N of codimension one in G such that T is induced from an irreducible S of N. By Mackey,

T=ind~(xS)

for every x in G. And by continuity of the restriction map and the fact that TIN is supported on the G-orbit G•S, we have x•SEV(6.N) for some x in G. Assuming inductively

/\ *;

that N ~n N is continuous at the identity representation, we may conclude by proper choice of

o

that the distance from the orbit

Q ( x •S) to 0 in ~:_

*

·is less than £. Now this distance is greater than the distance from Q(T) to 0 in

'J*,

and it follows that

1\ c;."'

G ~I /G is continuous at 1, completing our proof. QED

AN EXAMPLE

Let ~=~,

9

⁽⁰⁾ denote the exponential Lie algebra given by the nonzero basis relations [e₁,e₂]=e_{2 ,} [e₁,e₃]=-e_{3 ,} [e₂,e₃]=e_{4 ,} and let { e:} 1=₁ be a basis for

'd *

dual to { ei} 1=_{1 •} The nilradical

N=exp(Re₂+Re₃+Re_{4 )} of the group G=exp~ is isomorphic with the three dimensional Heisenberg group. For v*O, let Sv be the element of ~ associated to the functional of

n.

^The

isotropy group of

s

v is all of G, and

s

v extends to G. We denote by

s

the extension of

s

that corresponds to ve

*

₄ in

v v

'J*.

All the extensions of

s

_v are of the form T a,v =S v"Xa where x a is the · character .of G, equa 1 to 1 on N, with function a 1 ae* 1 , aER. The orbit Q

a, v of T a,v is seen to be the

hyperboloid

with a distance 12ii'V from zero, whenever O<v<a. Hence a sequence ^[Q _{t a v}

n' n

of such orbits may converge to 0 even if and ^a ++co ¹

n let e .·g.

v

_n ⁼¹

/n

² _'

representations, f T · )

l a: ' v . n n ·.

and

an=n. Then the two sequences of

{sv },

both converge to lG by

n

continuity of the Kirillov corresp6ndence. Still the sequence

v +0 n

of characters is unbounded. This answers in the negative a question raised at the end of [6].

REFERENCES

1. Auslander, L., Kostant, B., Polarizations and unitary

representations of solvable Lie groups, Inventiones math. 14 {1971), 255-354.

2. Baggett, L., Kleppner, A., Multiplier representations of abelian groups, J.Funct.Anal., 14 (1973), 299-324.

3. Boidol, J., ~-Regularity of Exponential Lie Groups, Inventiones math. 56 (1980), 231-238.

4. Brown, I.D., Dual topology of a nilpotent Lie group, Ann.Sci.

Ecole Norm. Sup., (4) 6 ( 1973), 407-411 .

....

5. Dixmier, J., Les C -algebres et leurs representations, Gauthier-Villars, Paris 1964.

6. Fujiwara ^I H. ^I Sur le. dual d ^I un groupe de Lie resoluble exponentiel, J.Math.Society Japan, 36 (1984), 629-636.

7. Joy, K.I.,

A

description of the topology on the dual space of a nilpotent Lie group, Pac.J.Math.,. 112 (1984), 135-139.

8. Kirillov, A.A., Unitary representations of nilpotent Lie groups, Usp. Math.Nauk 17 (1962), 57-110 = Russian Math.

Surveys, 17 (1962), 53-104.

9. Mackey, G.W., Unitary representations of group extensions, Acta Math., 99 (1958), 265-311.

10. 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, 93-94 =Functional Anal.Appl. 3

( 1 969) ¹ 340-342 ^o

11. Pukanszky, L., On unitary representations of exponential groups, J.Func.Anal., ·2 (1968), 73-113.