*-Regularity of exponential Lie groups

Ivl ve~l tlO ~le$

Inventiones math. 56, 231-238 (1980)

mathematicae ~ by Springer-Verlag 1980

,-Regularity of Exponential Lie Groups J. Boidol UniversitSt Bielefeld, Fakult/it fiir Mathematik, UniversitMsstrasse 1, D-4800 Bielefeld I, Federal Republic of Germany

Let ~r be a (*-semisimple) Banach-*-algebra and let C*(,~) be its C*-hull. Let Prim C*(.~t') be the primitive ideal space of C*(,4) and P r i m , , ~ the space of kernels of topological irreducible *-representations of ~_J in Hilbert space, both equipped with the Jacobson topology. Then we have a canonical mapping

7/: Prim C * ( , ~ ) ~ Prim,,~' ..r

J c~'

which is continuous and surjective. We want to call a (,-semisimple) Banach-*algebra ~J *-regular, if 7j is even a homeomorphism. Similarly we call a locally compact group G *-regular if its group algebra L~(G) is ,-regular. In [2] the problem was posed to determine the class [7j] of all *-regular locally compact groups. The main results were (A) I f G is *-regular, then it is amenable. (B) All groups G with polynomial growth are *-regular. (C) All semidirect products G =-A ~ N with abelian separable A and N such that IV/A is a To-space are *-regular. Furthermore it was shown that all connected simply connected solvable (real) Lie groups G with dirn G < 4, G 4=G4, 9(0) = exp 94.9 (0) in the notation of [1, p. 180] are *-regular. Then in [3] it was shown following a suggestion of D. Poguntke that G=G4,9(O ) is not *-regular. Thus there is exactly one connected simply connected solvable Lie group G with dim G <4 which is not ,-regular. In this paper we give a characterization of *-regular exponential Lie groups (3 ~ exp 9. Put for leg*, the real dual of 9, g ( f ) = {X e.qlf([X, 9])=0}, re(f)= g(f) + [.q, gl and m(f)~'= (~ r k=l

where (s

is the k-th term in the descending

central series of re(f). Our characterization is the following Theorem..q is *-regular if and only, !/'f(m(f) ~') = 0 for all fe9*.

0020-9910/80/0056/0231/$01.60

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J. Boidol

We recall that a locally compact group G is called symmetric if for all a~LI(G) a*a has real positive spectrum. As G4,~(0 ) is also the only nonsymmetric connected, simply connected solvable Lie group G with dim G < 4 (see [7]), one sees that symmetry is equivalent to .-regularity for connected solvable Lie groups G with dim G <4. Using the list of D. Poguntke of all nonsymmetric solvable Lie algebras g with dim g < 6 in [8] one can prove that this rests true for all connected solvable Lie groups G with dim G <6. Furthermore in [9] D. Poguntke shows using our criterion that for a big class of exponential Lie groups G = e x p g , namely those which admit Jordan decomposition in ad(g), .-regularity is equivalent to symmetry. Thus there is some hope that at least for exponential Lie groups .-regularity is equivalent to symmetry and investigations in .-regularity will help to solve the symmetry problem. This paper grew out of a part of the author's doctoral dissertation which was completed at the University of Bielefeld in 1979 under the direction of Professor Dr. Leptin. The author wishes to express his thanks to Prof. Leptin, Pro[ Poguntke and Dr. Ludwig for many fruitful conversations about the topic.

Notation. If 7c is a unitary representation of a locally compact group G, kern 7r will always denote the kernel of 7z considered as a representation of L~(G). w 1. Let G be a locally compact, a-compact group and let H be a closed subgroup of G. By A~ and A/t we denote the modules of G and H. Moreover for all h~H we define

(1)

A ~ . , ( h ) = (A,~(h)/A,~(17))

and

coshc,,,(h)= 89

1].

(2)

Then co(h)= cosh~.H is a symmetric weight on H and we can form the Beurling algebra Ll(H, co) with norm Mg l[o, --=.t Ig(~)[co(~)d~. If w is a function on G we define a function on H by (3)

P W = WII 1 " A G , I I ,

where w m is the restriction of w to H. Proposition 1. Let u, v~,~/ (G). Then there exists a positive constant C,.~, such thut f o r all f E L 1(G)

IIP(u*f*v)ll~

< C ....

Ilfll~.

Proof For all bereft(G), the function h ~ defined by h~ Therefore t ~ If(t)l ~I Ih(t- ~~)1 d~ /t

~ h(t()d~ is in ,~ff(G/H). n

,-Regularity of Exponential Lie Groups

233

is in LI(G) for all feLl(G) and he,;,~g(G). N o w by Fubini

j.f(t) j/,(t ' ~)d~ dt = .f .{.,+(t)h(t --~ ~)dt d~ = .f f, h(~)de. G

It

If G

1t

Thus f *hlH~L t (H) with

IIf*hinllz <.t" If(t) j" lh(t ~~)ld~ dt <= II Ihl~ G

tlfll~.

(4)

tt

Now let u, ce,:K(G), f~LI(G) and w = u * f * v . Then wi, and w~l are in LI(H) and -.1

(w~)D* = Wl.' 3<;. tl.

(5)

We get now by (4) and (5)

IlWlnA~.~t llo~<

.(Iw(h)l A~.~t(h)(A~;.11(h)+ Ac,.~t(h))dh 1f

= 3 tw(h)l dh + j Iw(b)A.s,~,l (h)l dh = II w~. II, + II,,% II, tl

tl

= IJu*f* t,l.]I, + IIv * * f * * u ~ l ] l ,

< (I v o Ix, I[u Ill + II lu*l~ ~. IIviii) Ilfll~. Proposition 2. Let G be a locally compact, a-compact group and let H be a closed subgroup. Let ~z be a unitary representation of H and p = i n d ~ the unitary

representation of G induced by ~z in the sense oJ Mackey and Blattner. Then kern p = {JeL 1(G)lP(u*f* v)ekern ~ for all u, ve,)f(G)}

where P is the mapping defined by (3). Proofi It is certainly sufficient to prove the proposition for cyclic 7r. Let ~ , Y ~ , the Hilbert space of re, be a cyclic vector for ~. Then . , ~ = ~ v* *u(h) A~,.5(h ) Qr(h) ~, ~) dh

n

defines an inner product on , ~ ( G ) and p is equivalent to the extension of left translation to the corresponding Hilbert space. Let fE,)V(G). By Fubini

(p(f) u, v),,~ - j" P(v* *f *u)(h) (~(h) ~, ~) dh.

(6)

H

S i n c e f - ~ P ( v * * f . u ) is continuous from I2(6) to LI(H,o)), we have Eq. (6) for all .f~L 1(G), u, w<~'(G). Translation of u and v by elements of H and the fact that is a cyclic vector for ~ gives the proposition. w 2. In this paragraph let g be an exponential Lie algebra. This means that g is real solvable with roots of the form ~p(x)(l +i~), where ~b is a real linear form on .q and eelR. Let for f e g * , the real dual of 9,

re(f) := .q(f) + [g, g]

(7)

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J. Boidol

where g ( f ) = {Xegl.f([X,g])=O}. Then m(f) depends only on (2f, the orbit o f f under the coadjoint representation of G, and we define for f2eg*/G, the orbit space of the coadjoint action of G on g*, m(f2) = m ( f )

(8)

where f is an arbitrary element of f2. Furthermore let (~ Cgkm(f)

m{f)~

(9)

k=I

where ~gm(f) is the k-th term in the descending central series of re(f), m(f) ~ is the smallest ideal a in re(f) such that m(f)/o is nilpotent. If b is a subalgebra of g we define on [9 Aq,~(X) = tr ad X - tr (ad X)l b.

(10)

Lemma. Let X e g ( f ) be generic for re(f). Let ml=m1(X ) be the Fitting-onecomponent and m o = mo(X ) the Fitting-null-component of(ad X)I,,Ij,). Then m(f) ~~= ml(X ) + [ml (X), mt(X)] = <mi(X)> ,

(11)

where (mz(X)) is the subalgebra of g generated by m1(X ). Proof Put f = m I + [m> m,]. Since [mo, mr] c_ml, we have [mo, 7] c [mo, ml ] + [mo, [m,, ml] ] c m, + [m,, [rno, mr] ] + [m,, Ira,, mo]]

cmx + [m,, mt] = L Furthermore Era> re(f)] = [m,, mo + ml] c_ m, + [m,, m,] = 7. This gives [m(f), 7] _ [mo, 7] + EmI, l ] c I. Since m , = [ X , ml] we have mlc__m(f) ~176 and it follows that [ is an ideal of m(f) contained in re(f) ~ and that [ = (mt(X)). One has m ( f ) = m o + ( m l ) , and m o is nilpotent because X is generic, so m ( f ) / ( m i ) is nilpotent and m(f) ~176 _ (ml). Proposition 3. Let g be exponential, f e g * with f(m(f)~)#O, Let p be a real polarization for f Then there exists a subalgebra D of g with p gl) such that

G, dg(f)) ~=o. Proof Let ( { 0 } = a o < a l < . . . < a l =g) be a Jordan-H61der sequence of the gmodule g. Let [~=ta+a~. Then p = I o < f ~ < . . . < [ t = g is a sequence of subalgebras of g which are g(f)-invariant, since g(f)_cp, g(f) acts on g/p and we claim that there exists X e g ( f ) such that a d X is not nilpotent modulo p.

,-Regularity of Exponential Lie Groups

235

Assume the contrary: T a k e X e g ( f ) generic for m(f). Let m~(X) be the Fitting-one-component and mo(X ) the Fitting-null-component of (adX)l,,,~j.). Then by (l 1) m(f)'~: = m,(X) + [m,(X), m,(X)] = (m1(X)~. Since m l ( X ) = a d X m l ( X ) and mo(X ) is nilpotent with Xemo(X), there exists k e n such that for allj>_k

m I (X) = m1(X ) + (ad X) J mo(X ) = (ad X) .i re(J). Since ad X is nilpotent m o d u l o p there exists k o >_ k such that for all j>= k o

(adX)Jm(f)~_p,

hence m l ( X ) = [ X , mdX)]c_[p,p ].

We conclude that

(m1(X)) = m(./)~' _ [p, p] c_ k e r n f which contradicts our assumption. N o w let X e g ( f ) such that a d X is not nilpotent m o d u l o p. Let i o be the largest index such that ad X operates with nontrivial eigenvalue e on [io+l/tio. We get that tr ad X - tr (ad X)1% = tr (ad X)I%+ 1 - tr (ad X)I% 4=0, since Reck4=0 also, by the fact that .q is exponential. Thus with h=[io the proposition is proved. Proposition 4. The set {m(f)~'~'[feg *} is a finite set oJ ideals (~f g.

Proof. re(f) is an ideal of 9 and m ( f y a characteristic ideal of re(f), hence the re(f) ~' are ideals of 9 for all f e g * . We prove that even the set M TM

= {a~l[g, 9 ] - a <.q} is finite. Let ~o~..... (p, be the roots of the 9-module g. Let a be an ideal with [9, 9 ] - a such that no root of 9 vanishes on a. Then a ~ = g~ by (11). Now M = M o w M 1 w . . . w M , where M o = { g ~} and M~ = { a ~ l [9, 9] ___a < g, ~0~(~) = 0}.

An induction a r g u m e n t gives that the sets M~ are finite, thus M is finite. w Let G = e x p g be an exponential Lie group. F o r fe.q* we denote by ~ ( f ) the unitary representation of G associated to f by the Kirillov mapping. Let

G,~ = {rr = rr(f)eG ^ ]re(f) nilpotent}. L e m m a 1. Let 9 be exponential, G = e x p g. Let {~i}~., = {~(f~)}i~J be a familiy in G~ and let ~Zo=~(fo)EG ^ with kern n~ _~kern )zo.

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J. Boidol

Then Gjoe{GfiliE~r

in g*/G.

Proof. Let n < ,q be the nilradical of g, let N = exp n. If l e g * and 7t = ~z(f), then nl,~' is weakly equivalent to {g-n(Ji,)Ig~G}, where n(,[i,)eN ~ is the irreducible representation associated for Ji-" But {g,Tt(/i,,)IgeG } is weakly equivalent to {~z(ad* (g)Jj.)l g eG} = {~(hl.)l h eGf}. Let now {~zi},~j= {=(J)}i~j be a family in G~' and ~zo=~Z(Jo)eG ~ such that kern =~g kern ~oi6,~

Then -]kern =i~,~~- kern ~Zo~N ieJr

and since N is regular as a nilpotent group, ~o,~ is weakly contained in the {rri~}i~ J. But then by the remarks above {lt(ad*(g)fol.)geG} is weakly contained in {rc(ad*(g)jli.)lgeG, i ~ J }. Since by Brown's theorem (see [4]) the Kirillov map is a homeomorphism from n*/N onto N A, we get Gfol" c_ i ~ Gj;I"

in n*.

We fix now a complement W of n in g and identify n* with W • in the canonical way. Then we get: G f o ~ n • q-Gfol, ~ n •

U GJlt,,~ U n•

icJ ~

is~

By [6, Prop. 1.1.], f+m(Gj)Zc_G~ for all feGJi and therefore:

But this gives GJoe{GfilieJ }- in g*/G. Lemma 2. Let ,q be exponential, G =expg, go
Proof. Let {al}i"=1 be a good sequence of subalgebras of g such that %_ 1=go. (See [1, Chap. IV, 4.3] for the definition.) Let Jl =.~,,. Then p = ~ ai(~) is a real n

1

i=1

polarization for f and pO = ~ ai(j ) is a real polarization for f o =Jigo" i=1

to and pO satisfy the Pukanszky condition and p = g ( f ) + p O . Let ~ o = r t ( f ~ be the irreducible unitary representation of G o associated to fo. Then the restriction map ~o--+q~lGodefines a unitary operator from ~ to ~f~o which intertwines rClG,, and ~o. Therefore ~tao is irreducible and X - ' X | is an injective mapping from (G/Go) ^ into G ^. Since G o is type I, this mapping is even a homeomorphism by [5, Prop. 6].

,-Regularity of Exponential Lie Groups

237

Theorem. Let G =exp.q be an exponential Lie group. Then G is *-regular if and

only !f f ( m ( f ) ~ ) = O fi~r all J'e.q*. Furthermore !f G is .-regular, then the Kirillor map is a homeomorphism and g*/G ~ G A _~ Prim C* (G) ~ Prim. L1(G) under the canonical mappings. Proof. a) We assume that there exists l e g * such that f ( m ( f ) ' ) : f O . Then after proposition 3 there exists a real polarization p for ,s satisfying the Pukanszky condition, and a subalgebra b of ~ with p_c b such that Aq.~(.q(f))+0. Let zleq* be an arbitrary extension of A~,~ to .q such that A([.q,.q])=0. Let ,q0=kernA. Then .% is a one codimensional ideal of .q and .qo+.q(f)=q. Let G o = e x p G o, H = e x p b , P = e x p p . Zf the unitary character of P defined by f The real character defined by -Aa, tj on H is A(~,/4, which is equal to the restriction to H of the real character 6 defined by - A on G. We set H 0 =kern da.,,, ~ ' = i n d f Zr and ~ = 7 ~ ( f ) = i n d ~ z r = i n d ~ f a . Identifying (G/Go) ^ ~(H/Ho) ~ ~-IR we consider the subspace S={Z|

A}

of G ~,

which is after Lemma 2 in a canonical way homeomorphic to IR. As ZQzE ind~ ~ @ a we get by Proposition 1 : kern Z | 7~= {./ELi (G)I .[ P ( u . f . 0(h) z(h) a(h) dh = 0 for all u. r e K (G)} /4

={feL~(G)l ,t" .t" P(u*f*l~)(hho)a(hho)dhoZ(li)dl~=O /4//40 tto

for all u, re,~(G)}

= {feL~ (G)l .[ P,~.,:.~.,(f)(l;) z(li) dl;=O H/Ho

for all u, vc,X/(G), ~. r/eJq~}, where Ps

.[ P ( u . f .v)(hho) dh o. Finally we get Ho

kern 7~| rc = {J'eL l (G){ [P~,, r

* (Z) = 0

for all u, ve#Z(G). ~, rle~:#~}. But after Proposition 1 P~,,r This implies that [P~, r ~' is analytic. Now let J# c l R be a set which is closed and which has a non empty interior. Then

L(G)I[P,I.,.r

Okernz|

Zeta

(z)=O for all Xe'~//,u, r e , ~ ( G ) , ~,~/e,~}

= { f e L ~(G)I [P,[,.. ~.,(f)] ~ (Z)= 0 for all ze(H/Ho)A. --

0

ZelG/Go) A

kern Z | ~-

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J. Boidol

It follows that s~:{x| is dense in S with respect to the topology of P r i m , L1(G) but closed in S with respect to the t o p o l o g y of G ~ ~ Prim C* (Lt (G)). Therefore the canonical m a p p i n g t/,: Prim C,(L,(G))_,Prim, Ll(G) is no hom e o m o r p h i s m and G is not ,-regular. b) F o r dim g = 1 the theorem is certainly true for in this case G ~ I R . We assume that the theorem has been proved for exponential Lie groups G = e x p g with dim g < n . Let now g be an exponential Lie algebra with dim .q=n such that for all l e g * , f(m(f)~176 Let {TTi}ie~:{Tc(fi)}ie, ~ be a family of irreducible unitary representations of G = e x p g and rt 0 = =(fo)eG ~ such that (~ kern =z _c kern rt o we show that fo is contained in the closure of the union of the orbits of the .~. Let {a0, % .... , % } = { m ( f F ~

*}

(see Prop. 4) and A i = e x p a / .

We assume that the indices are chosen such that a 0 = {0}, a i :4={0} for i + 0. Since P r i m , LI(G) is a topological space we can first assume that all f~ are of the same kind, i.e. m(f~)~=ar for all i e J , where re{0, 1..... k}. If r : # 0 this means that all rci and then also rt 0 vanish on Ar and we are able to use our induction hypothesis for G/A r. If r = 0 we are in the case of L e m m a 1. Thus we get that the canonical m a p p i n g P r i m , LI(G)--+ g*/G is continuous. Since the Kirillov m a p p i n g is continuous G is ,-regular and since ~ : Prim C*(G)~ P r i m , L~(G) is continuous the Kirillov m a p p i n g is a h o m e o m o r p h i s m . Finally, since G is type 1

g*/G ~- G A ~ Prim C* (G) ~ P r i m , L1(G) and the theorem is proved.

References I. Bernat, P., Conze, N.: Repr6sentations des groupes de Lie r~solubles. Paris: Dunod 1972 2. Boidol, J., Leptin, H., Schiirmann, J., Vahle, D.: RS.ume primitiver ldeale von Gruppenalgebren. Math. Ann. 236, 1-13 (1978) 3. Boidol, J.: On a regularity condition for group algebras of non abelian locally compact groups. Preprint, Bielefeld 1978 4. Brown, I.: Dual topology of a nilpotent Lie group. Ann. Sci. l~cole Norm. Sup., 4 ~ sdrie, t. 6, 407411 (1973) 5. Dixmier, J.: Bicontinuit6 dans la m6thode du petit groupe de Mackey. Bull. Soc. Math. 2 e s&ie. 97, 233-240 (1973) 6. Duflo, M.: Caract6res des groupes et des alg6bres de Lie r6solubles. Ann. Sci. l~cole Norm. Sup., 4" s6rie, t. 3, 23-74 (1970) 7. Leptin, H., Poguntke, D.: Symmetry and nonsymmetry for locally compact groups. Functional Analysis, in press (1980) 8. Poguntke, D.: Nicht symmetrische sechsdimensionale Liesche Gruppen. Journal Reine Angew. Math. 306, 154-176 (1979) 9. Poguntke, D.: Symmetry and nonsymmetry for a class of exponential Lie groups. Journal Reine Angew. Math., in press (1980) Received September 14, 1979