C OMPOSITIO M ATHEMATICA
N IELS V IGAND P EDERSEN
Lie groups with smooth characters and with smooth semicharacters
Compositio Mathematica, tome 48, n
o2 (1983), p. 185-208
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185
LIE GROUPS WITH SMOOTH CHARACTERS AND WITH SMOOTH SEMICHARACTERS
Niels
Vigand
Pedersen*© 1983 Martinus Nijhoff Publishers - The Hague Printed in the Netherlands
Introduction and main results
Let G be a
connected, simply
connected Lie group. In a recent paperwe showed that if G is solvable and of
type
R(i.e.
all its roots arepurely imaginary),
then all characters of G aresmooth,
thatis, they give
rise todistributions on the group
([19]
Theorem4.1.7,
p.239).
In thespecial
case where G is
nilpotent
this result is due to Dixmier([7]).
If G issemisimple
the same result holds as shownby
Harish-Chandra([13]) (At
thispoint
we would like to remind the reader that whendealing
with solvable
(or
moregeneral)
Lie groups the notion of a character ismore
complicated
than e.g. fornilpotent
orsemisimple
groups; this is due to the fact that solvable Lie groups are notnecessarily type
7(for
characters in
general,
cf. e.g.[9] §6;
for characters on connected Lie groups, cf.[25])).
The limits for the class of Lie groups for which such aresult
(i.e.
the smoothness of allcharacters)
can hold was setby Pukanszky ([26], [27],
cf. also the paper[18]
ofMoore, Rosenberg):
Heshowed that if all normal
representations (cf. [25])
are GCCR(cf. [27],
where this notion is
defined;
this can berephrased by saying
that thenormal
representation
has adensely
definedcharacter,
cf. Lemma1.2.1)
then G must be the direct
product
of a connectedsemisimple
Lie group and of a connected Lie group whose radical iscocompact
andtype
R.This
implies
inparticular
that groups for which all characters aresmooth must be of this form. One of the purposes of this paper is to extend our result of smoothness of all characters to this
largest possible
class of groups.
Specifically
we have:* Supported by a fellowship from the University of Copenhagen and by Det Natur- videnskabelige Forskningsrâd, Denmark. This work was performed while the author was a visitor at the Department of Mathematics, University of California, Berkeley.
0010-437X/83/02/0185-24$0.20
THEOREM 1: Let G be a
connected, simply
connected Lie group. Thefollowing
conditions areequivalent:
(i) Any
normalrepresentation of
G has a smooth character.(ii) Any
normalrepresentation of
G has adensely defined
character(or, equivalently,
isGCCR, cf.
Lemma1.2.1).
(iii) Prim(G)
is aTl topological
space.(iv)
G is the directproduct of
aconnected, simply
connectedsemisimple
Lie group and a
connected, simply
connected Lie group whose radical is cocompact and type R.The
equivalence
of(ii), (iii)
and(iv)
is due toPukanszky ([26], [27]
Proposition 3).
Theequivalence
of(iii)
and(iv)
was alsoproved by
Moore and
Rosenberg ([18]
Theorem1);
cf. also the papers of Moore[1], Chap.
V andLipsman [15].
In
[19]
we also showed that for anarbitrary connected, simply
con-nected solvable Lie group it is
possible
to associate a semicentral distri- bution to any normalrepresentation
of G([19]
Theorem4.1.1,
p.232).
Our results in
[19]
were formulatedby
aid of the notion of a semichar- acter, defined to be a certainrelatively
invariantweight
on the groupC*-algebra.
We said that a semicharacter is smooth if itgives
rise to adistribution on the group
(in
which case the distribution is a semicentraldistribution).
The main purpose of this paper is to confine the class of Lie groups for which all normalrepresentations
have a smooth semi- character.Specifically
we prove:THEOREM 2: Let G be a
connected, simply
connected Lie group. Thefollowing
conditions areequivalent:
(i) Any
normalrepresentation of
G has a smooth semicharacter.(ii) Any
normalrepresentation of
G has adensely defined
semi-character.
(iii) Any
normalrepresentation of
G is either GCCR or isinduced from
a normal GCCR
representation of
aconnected,
normalsubgroup of
codi-mension one
(or both).
(iv)
G is the directproduct of a connected, simply
connectedsemisimple
Lie group and a
connected, simply
connected Lie group whose radical is cocompact.At this
point
it will beappropriate
to mention thefollowing
newfeature
arising by
the introduction of semicentral distributions for thedescription
ofrepresentations
of connected Lie groups: If v is a semicen- tral distribution ofpositive type
there is associated arepresentation Âv
to v
(via
theGNS-construction).
Let us assume that thisrepresentation
is factorial. It is then a non-trivial
problem
whether therepresentation Àv
is normal and even whether03BB03BD generates
a semifinite factor.This, however,
we settled in the affirmative in a recent paper([21]).
Moreprecisely
we showed that iff
is a semicharacter on a connected Lie group then therepresentation 03BBf
associated tof (via
the GNS-con-struction)
isactually
normal. We can thus summarize the main features of our results as follows:Starting
out with a semicentral distribution ofpositive type
which is ’extremal’(i.e. Àv
is a factorrepresentation)
leadsus to a normal
representation,
and all normalrepresentations
can beobtained in this fashion
precisely
if the group satisfies condition(iv)
inTheorem 2.
As it will be seen from Theorem
1,
aconnected, simply
connected Liegroup
G,
fails to be GCCR if at least one of thefollowing
twopheno-
mena occurs: 1. the radical of G is not of
type
R or 2. G has non-compact semisimple subgroup acting non-trivially
on the radical.From Theorem 2 it is seen that if we use the broader notion of a semi- character
only
thephenomenon
2. cannot beeffectively
dealt with. This makes us venture thefollowing
remark: If 03C0 is a normalrepresentation
of a
separable locally compact
group and if 03C0 is not GCCR there seems to be two distinct ways in which can fail to be so, and these areclassified
according
to whether or not there exists a continuous homo-morphism
x : G -Rt ,
such that 03C0 has adensely
defined semicharacter(we
shall resist thetemptation
to call n SGCCR(= semi-GCCR)
if sucha x
exists).
For
unexplained
notation we refer to the notational conventionsadopted
in[19]
and[21].
1 would like to thank the
Department
ofMathematics, University
ofCalifornia, Berkeley,
where this work wasperformed,
for itshospitality during
mystay
in1979/80.
1. Preliminaries
1.1. In this section we shall make a few remarks about smooth semi- traces. This will be useful in the
following.
Let G be a Lie group and let x : G ~
Rt
be a continuous homomor-phism.
Recall that ax-semitrace f
on G(cf. [19]
Definition2.1.5)
iscalled smooth if
C~c(G)
cm f
and if ~ ~f(~) : C~c(G) ~
C is a distribu-tion on G
([19]
Definition2.3.4).
We have the
following
converse of[19] Proposition
2.3.5:LEMMA 1.1.1:
Let f be a
smoothx-semitrace
on G. Then there existsm~N,
such thatCm(G)
cmi.
PROOF: Since a distribution is
locally
of finite order there exists anopen
neighborhood U
of theidentity
inG,
such that the restriction off
to
C~c(G, U)
is continuous in theC’(G)-topology
for some m E N. Let Vbe an open,
symmetric neighborhood
of theidentity
inG,
such thatY2
c U. Let qJ EC’(G, V).
There exists a sequence qJn EC~c(G, V),
suchthat qJn --+ 9 in the
Cm(G)-topology.
Inparticular
~n - ~m ~ 0 for m, n - + cc, in theCm(G)-topology,
and therefore(qJn - qJm)*
*(qJn - ~m) ~
0 inthe
Cm(G)-topology,
and thisimplies
thatf«9n - qJm)*
*(9n - ~m)) ~
0for m, n - + oo, since the restriction of
f
toC~c(G, V)
is continuous in theCm(G)-topology.
But this shows that(jJn
Enf/n0f
cH f
is aCauchy-
sequence, hence in
particular ~~n~2
=f(~*n
*qJn)
isconvergent.
But since also ~n ~ 9 inC*(G)
we have thatf(~*
*ç) ~
liminff(~*n
*~n)
+ oo, and this shows thatC’(G, V)
cnf.
With
m, V
asabove,
let m’ EN, t/!, 03C80
ECm(G, V)
as in[19]
Lemma2.3.6,
p. 206. From the formula in[19],
p.206, bottom,
it then follows thatC2m’c(G)
cm f.
This proves the lemma.Let now X be a Borel space and let p be a measure on X.
Moreover, let 03BE
-f03BE, 03BE
EX,
be afamily
ofx-semitraces
onG,
suchthat 03BE ~ f4(x)
isborel for all x ~
C*(G)+.
Assume that there isgiven
ax-semitrace f
onG,
such that the formulaholds for all x ~
C*(G) +.
We then have:LEMMA 1.1.2:
If f
issmooth, then f03BE is
smoothfor
almostall 03BE
~ X .PROOF: Choose m ~
N,
such thatCmc(G)
cm f (Lemma 1.1.1)
andchoose m’ ~ N
and 03C80 ~ Cm(G)
as in[19]
Lemma2.3.6.,
p. 206.Arguing
as in the
proof
of[19] Proposition 2.3.5,
pp. 206-207 we findthat f03BE
issmooth for
those 03BE
for which03C8,03C80~nf03BE,
and sinceclearly 03C8,03C80~nf03BE
foralmost
all j
EX,
we haveproved
the lemma.Let N be a
closed,
normalsubgroup
of G andset ~ =
- ~ · (~G/N o cG/N)-1.
Assume thatf
is a(G, ~)-semitrace
on N(such
thatin
particular f
is a semitrace on N withmultiplier X 1 N,
cf.[ 19] 2.1).
Theinduced
semitrace = indN~TGf
is then well-defined([19]
Definition2.1.2).
is ax-semitrace
on G. With this notation we have:PROPOSITION 1.1.3:
If f is smooth,
then= indNiGf
is smooth.Moreover, 1«p)
=f (~|N) for all 9 c- C ’ (G).
PROOF: Since
f
is smooth there exists m~ N such thatC’(G)
cm f (Lemma 1.1.1).
Inparticular 1«p *~)
=f(~* * qJIN)
for ç EC’(G) ([21]
Proposition 2.1.3),
and thusC’(G)
cnf.
It follows from[21] Proposit-
ion
2.3.5,
p. 206that 1
is smooth.Moreover, by polarisation
weget
that(~)
=f(~|N)
for all (p EC’(G)
*Cmc(G) ~ C’(G) (cf. [2],
p.252, top).
This proves the
proposition.
1.2. Let G be a
separable locally compact
group. In[27],
p. 40Pukanszky
defines what if means that a factorrepresentation
of G isgeneralized completely
continuous(GCCR).
Thefollowing
lemmaclarifies the situation:
LEMMA 1.2.1: A normal
representation
TCof
G is GCCRif
andonly if
ithas a
densely defined
character.PROOF: Lest 91 =
x(G)"
andlet 0
be afaithful, normal,
semifinite traceon 21. Then
f = ~o03C0|C*(G)+
is a character for 03C0. In[27],
p. 40 it wasshown that the linear span
C(21)
of the set of elementsT ~21+,
such that~(I - EÂ)
+ oo for all 03BB >0,
where T= f ô ÀdEÂ
is thespectral
re-solution of
T,
is anormclosed,
two-sided ideal in 21. Now it iseasily
seenthat if
T ~ m+~ ,
thenT~C(21)+,
and thereforeiff (N) (normclosure)
is con-tained in
C(21). Conversely,
if TEC(W) +,
it was shown loc. cit. that T can beapproximated
with elements fromm+~,
and thereforem~(N)
C(çX).
We have thus seen that 7r is GCCR if andonly
ifTC( C*( G»)
cm~(N).
But this isequivalent
to the characterf being densely
defined
([21]
Lemma3.4.2).
2. Proof of Theorem
2, (i)
~(ii)
This is trivial.
3. Proof of Theorem
2, (ii)
~(iii)
Assume that
(ii)
is satisfied and that the normalrepresentation x
of G isnot GCCR. Let x : G ~
Rt
be a continuoushomomorphism
and let03C9 be a
x-relatively
invariantweight
on N =x(G)",
such thatf on 1 C*(G)+
is adensely
definedx-semicharacter
for n. Inparticular x Q
1. It follows from[20]
Theorem 6.2 that there exists a normal re-presentation
rco ofGo
=ker x,
such that x isequivalent
toindGo
i G7ro.Moreover there exists a character
fo
for xo, such thatwhere
f,
is the(G, x)-character
onGo
withindGo i Gfl
=f (cf. [21]
Theorem
3.3.1).
Now sincef
isdensely
defined it follows thatf,
isdensely
defined(cf.
loc.cit.)
and thereforefo
isdensely
defined. But thenxo is GCCR
(cf. 1.2).
SinceGo
is aconnected,
normalsubgroup
of codi-mension one we have
proved
that(ii)
~(iii).
4. Proof of Theorem
2, (iii)
~(iv)
Assume that the
connected, simply
connected Lie group G is not of the form described in(iv).
We shall exhibit a normalrepresentation
ofG,
which does not have theproperty
described in(iii).
4.1. For this it will sufHce to exhibit a
connected,
normalsubgroup B
ofG,
such thatG
=G/B
has anirreducible,
normalrepresentation
iî withthe
property
that à is neither CCR nor is induced from aCCR-represen-
tation of a
connected,
normalsubgroup
of codimension one. Infact,
if iîis such a
representation,
define therepresentation 03C0
of Gby 03C0(s)
=(),
s = sB. Then 03C0 is an irreducible
representation
ofG,
which isnormal,
but not CCR. Assume then that 03C0 is induced from a normal represen- tation rco of the connected normal
subgroup Go
whose codimension isone.
Clearly
03C00 isirreducible,
and the stabilizer of 03C00 in G isGo,
sinceotherwise 03C0 =
indGo
~ Gp0 would not be irreducible. Now the restriction of 03C0 toGo
isequivalent
to(the
notationexplains itself)
and thisdecomposition
iscentral,
since 03C00 is GCR and since the stabilizer of xo isGo.
It follows from this that all elements in B leaves 03C00 invariant(since 03C0(b)
=I,
b EB,
and so03C0(b)
in-duces the trivial
automorphism
in the von Neumannalgebra generated by
therepresentation (*)).
But then B cGo
and we can define the re-presentation no
onGo
=GOIB by 7To(.s)
=7r,(s),
and we have n ==
indGo ~0
andGo
is aconnected,
normalsubgroup
inG
of codimen- sion one.Therefore, by assumption, no
is notCCR,
and so 03C00 is not CCR. This shows that it sufHces to exhibit a B as described.4.2. The rest of Sect. 4 will
inevitably
have a lot in common with[1],
pp.
171-174, [15],
pp.744-747, [18],
pp. 208-211 and[27],
pp.43-45,
to which papers we at certain
points
refer for further details.4.3. We first define some
special
groups which will be of interest in thefollowing.
Let S be a
connected, simply connected, non-compact, (real)
semi-simple
Lie group with Liealgebra
s.First,
let 6 be alocally
faithful(i.e.
the differential du isfaithful)
irre-ducible
representation
of S on Rn. We define the groupSi(J)
to be thesemidirect
product
RnxsS,
where S acts in Rn via 03C3. We defineS2(u)
tobe the semidirect
product
Rns (S
xR),
where the directproduct
S x Racts in Rn
by (s, t)x
=etu(s)x,
SEG,
t ER,
x E Rn.Let next 6 be a
locally faithful,
irreduciblerepresentation
of S on C".We define the group
S3(U)
to be the semidirectproduct
C"x(S x C),
where the direct
product
S x C acts on Cnby (s, t)x
=etu(s)x, s E S,
t EC,
X E Cn.
Finally
we defineS(a, y)
for ,il ~ R to be the semidirectproduct
en
x s (S
xR),
where the directproduct
S x R acts in C"by (s, t)x
== e(03BC+i)t03C3(s)x, SES, t c- R,
x E C".The groups
Sj(u), j
=1, 2, 3, S(6, ,u), ,u
eR,
areclearly connected, simply
connected Lie groups.
For the group G =
S1(03C3)
we definesubgroups
N = Rn s{0}
and A ==
{e},
and for G =S2(03C3)
we set N = Rns({e}
x{0})
and A ==
{0} s({e}
xR). Similarly
for the otherspecial
groups. In thisfashion,
if G is one of the groupsSj(03C3), j
=1, 2, 3, 8(u, J.l), J.l
ER,
then N is the nilradical of G and G = NAS. The radical R of G is AN.We say that a Lie
algebra
has nosemisimple
directfactors,
if its max-imal
semisimple ideal,
which is a directfactor,
is trivial(cf. [27],
p.43, top).
LEMMA 4.3.1:
If G is
oneof the
groupsSj(a),j
=1, 2, 3, S(6,,u),
ye R,
wehave
(g being
the Liealgebra of G): (i)
g has nosemisimple direct factors, (ii)
the radical rof
g is not cocompact,(iii)
the nilradical tt is a minimal proper ideal and(iv)
the radical r hasprecisely
one real root orprecisely
two
complex conjugate
roots.Conversely, if
G is aconnected, simply
con-nected Lie group
satisfying
theproperties (i)-(iv),
then G isisomorphic
toone
of
the groupsSj(03C3), j
=1, 2, 3, S(,7, IÀ),
03BC ~R.PROOF: The
special
groupsclearly
have theproperties (i)-(iv).
Assumeconversely
that r has twocomplex conjugate
roots (P = çi +i~2
andip
= ~1 -i~2,
such that ~1 and ~2 arelinearly independent,
and let usshow that G is
isomorhpic
toS3(u).
The other cases are treated in afashion similar to this one. Let s be a Levi
subalgebra
in g and let S bethe
analytic subgroup corresponding
to s. Since therepresentation
s ~
Ad(s) r
issemisimple
and since n is an invariantsubspace
under thisrepresentation,
there exists asupplementary subspace
a to n in r, such that a isAd(s)-invariant.
Since[r, s]
c n it follows that[a, s]
= 0. Sincen =
ker ç
=ker çi n ker 92
and since ~1and ~2
arelinearly indepen-
dent we have that dim a = 2. Pick
X1, X2 ~a with ~j, Xk> = 03B4jk, j, k
=
1, 2.
Then set V ={Z ~ n|ad X(Z) = ~(X)Z
for all X Erl.
Then V isa nonzero
subspace
in nc and it is invariant under G(since
a and scommute)
and since n is a minimal ideal it follows that nc = VE9 V
For all Z E V we have inparticular ad X1 (Z)
= Z andad X2(Z) = iZ,
from whichad ([Xl, X2]) = [ad Xl,
adX2]
= 0. But then[Xl, X2]
= 0. Lete1j
+
ie2j, j = 1, ..., p,
be a basis in V. Thene1j, ej2, i = 1, ..., p,
is a basis in n.Define the map n ~ n
by x 1 e i
+ ... +xne1n
+y1e21
+ ... +yne2n ~ (x1
+iy1,
..., xn +iyn) = (z1, ... , zn) =
z.Identifying
n with en in thisway we find
Ad(exp(t1X1
+t2X2))z
= eh+it2Z.
Let 6 be the represen- tation of S on ngiven by
s ~Ad(s)|n.
Since commutes withad Xx2|n
and since ad
X2
ismultiplication by
i onC",
it follows that the represen- tationactually
arises from arepresentation 03C3
on thecomplex
vectorspace C". The
irreducibility
and the local faithfulness of u are clear. We have thus shown that G isisomorphic
toS3(U).
For a
proof
of thefollowing lemma,
cf.[27], proof
of lemma23, Ad (A) (b),
pp. 43-44.LEMMA 4.3.2: The groups
Sj(03C3), j = 1, 2, 3, S(03C3, y), y
ElÉ,
are all type I.LEMMA 4.3.3: None
of the
groupsSj(03C3), j = 1, 2, 3, S(6,,u),
03BC ER,
havethe property
(iii)
in Theorem 2.PROOF: Let G be one of the groups in
question.
Recall that G =NAS,
and set K = AS. Let s, be asubalgebra
in 5,isomorphic
tosl(2, R),
andlet u’ be the
representation contragredient
to 6. Letf ~ n’, f ~ 0,
and letX ~ s1, X ~ 0,
such that6’(X)f
=03BBf,
where is anegative
real number.Let
then X
be the character on Ngiven by x(exp Y)
= expif, Y>, Yen,
and letK.
be the stabilizer of x in K. There exists a character 9 ofGx
=KxN,
such that~|K~ ~
1 and such that(p 1 N
= x.By Mackey- theory
x =indGx ~G~
is an irreduciblerepresentation
of G. Now(exp tX)~(exp(Y)
= expi03C3’(exp tX)f, Y)
= expi(e03BBtf, Y),
and thismeans that
(exp tX)x corresponds
to the functionale03BBtf,
and therefore(exp tX)~ ~
1 for t ~ + oo. SincesK~s-1
=Ksx
and since the stabilizer of(exp tX)X
is the same as that of x we see that exp tX normalizesK.
and that
(exp tX)~
extends(exp tX)~
on N and that(exp tX)~|K~
~ 1.It follows that
(exptX)~ ~
1 for t ~ + 00. Let us then show that7T is not CCR
(cf.
the references mentioned in4.2):
We have03C0 ~
(exptX)03C0 ~ indGx ~G(exptX)~ ~ indG~~G1.
If ker rc was a maximalideal we would have that
ker(indG~~G1)
= ker n and thusker(1|N)
=
ker(nIN),
which is false. We next show that 03C0 is not induced from aCCR-representation
of a connectedsubgroup
of codimension one. So letGo
be such asubgroup
and let 03C00 be an irreduciblerepresentation
ofGo,
such that 03C0 =
indGo~G03C00.
We have to show that 03C00 is not CCR. SinceSN c
Go
cG, Go
isisomorphic
to one of the groupsSj(u), j
=1, 2, 3,
with nilradicalisé
SN,
so SN isregularly
imbedded inGo (cf. [27] proof
of Lemma 23 Ad
(A) (b),
pp.43-44).
Therefore we can assume without loss ofgenerality
that 03C00 lives on the orbitGo ’
x inlV.
There existsby Mackey-theory
a character ~1 on(GO),,
such that~1|N
= x and such that xo =ind(G)~~Go~1.
But the 03C0 =ind(Go)~
~G~1which, by
the irre-ducibility
of 7r,implies
that(GO),
=Gx
and that ~1 = 9. We can thenargue
precisely
as above to show that 03C00 is not CCR. This ends theproof
of the lemma.LEMMA 4.3.4: Assume that G is a
connected, simply
connected Lie group notsatisfying property (iv)
in Theorem 2. Then G has aquotient
isomor-phic
to oneof
groups.Sj(u), j
=1, 2, 3, S(03C3,03BC),
03BC E R.PROOF:
Imitating
theprocedure
in[27] proof
of Lemma23,
Ad(B) (a),
p. 45 and Ad(A) (a),
p. 43 one can show that the Liealgebra
g of G has aquotient satisfying
theproperties (i)-(iv)
in Lemma4.3.1,
and theresult then follows from this lemma.
It follows from Lemma 4.3.4 and Lemma 4.3.3 that if a
connected, simply
connected Lie group G fails tosatisfy property (iv)
in Theorem2,
then G has aconnected, simply
connectedtype
1quotient
notsatisfying
the
property (iii)
in Theorem 2.Bearing
in mind our remarks in 4.1 wehave then shown that G does not have
property (iii)
in Theorem 2. Thisends the
proof
of Theorem2, (iii)
~(iv).
5. Proof of Theorem
2, (iv)
~(i)
We do this in several
steps.
It is a routine matter(using [13])
toreduce to the case where G
actually
hascocompact
radical.5.1.
So,
let G be aconnected, simply
connected Lie group with Lie al-gebra
g whose radical iscocompact.
We let r,b,
n denote theradical,
the nilradical(=
maximalnilpotent ideal)
and the nilradicalisé(= [g, g]
+
g), respectively,
and letR, H,
N denote thecorresponding closed,
con-nected,
normalsubgroups.
5.2.
By
Ado’s theorem([6]
Théorème5,
p.153)
we can without loss ofgenerality
assume that g is a Liesubalgebra
of the Liealgebra gl(V)
ofendomorphisms
of some real finite dimensionalspace V,
in such amanner that the
nilradical 4
of g consists ofnilpotent endomorphisms.
But
then 4
isalgebraic ([6] Proposition 14,
p.123)
and therefore also nis
algebraic ([10]
1.2.(ii) Lemma,
p. 424 and[6] Proposition 19,
p.129).
Let g
be thealgebraic
closure of g. Then[g, g] = [g, g]
and r,1),
n areideals
in g ([5]
Théorème13,
p.173).
Let f be the radical ofg.
Then f isalgebraic, and,
infact,
r is thealgebraic
closure of r([6] Proposition 19,
p.
129).
We can write f = a~
m, where m is the ideal ofnilpotent
ele-ments in f and where a is an
algebraic
abeliansubalgebra of g ([6]
Proposition 20,
p.130). Also g
=~ s and
g = r(B
s, where s is a Levisubalgebra
of g, and a and 5 can be chosen such that[a, s]
= 0([6]
Proposition 5,
p.144). By [21],
p. 477 b we can write a as a direct sumof
abelian, algebraic algebras
t andf,
such that all elements in t havepurely imaginary eigenvalues
and such that all elements in f have realeigenvalues.
Let
61
be theanalytic subgroup
ofGL(V) corresponding
tog. Gl
isthe connected
component
of analgebraic
group inGL(V);
inparticular
G 1
is closed. LetG
be the universalcovering
group of1.
We then have H c N c G cG,
all asclosed,
normalsubgroups,
and[G, Û] = [G, G].
By [24] Theorem,
p. 379 N and H are bothregularly
imbedded inG.
Let
T,
K and M be theanalytic subgroups
inG1 corresponding
tot,
and m,
respectively.
T and K are the connectedcomponents
of the id-entity
ofalgebraic
groups, hence closed. T isclearly
acompact,
con- nected Lie group and K is an abelian connected andsimply
connectedLie group. Set e = f (D m. e is a solvable Lie
subalgebra
of§
with realroots. Set E = KM. E is a
closed, connected, simply
connected sub- group inGL(V)
with Liealgebra
e(in fact, choosing
a properbasis,
e isa
subalgebra
of the solvable Liealgebra
of uppertriangular matrices),
soE is an
exponential, connected, simply
connected solvable Liealgebra
with real roots.
5.3.
Using
the notation fromabove, let 03C0 ~
and setN(03C0) = {03C1 ~ |03C1|H
lives on the orbit N .
03C0}.
LEMMA 5.3.1:
9(n)
is countable andindh i 17r
isquasi-equivalent
to@p6N(7T)P’
PROOF: Let
N03C0
be the stabilizer of 03C0 in N. For the first assertion it suffices to show([16]
Theorem8.1,
p.297)
that the set of elements in Nwhose restriction to H is a
multiple
of 03C0 is countable. But this isclear,
since this set can be identified with a subset of the dual of a certain
separable, compact
group(namely
an extension ofN03C0/H by
the circlegroup, cf.
[16]).
ThatindHil7l
isquasi-equivalent
to~03C1~(03C0)03C1
is con-tained in
[14]
Lemma 4.2(using [16]
Theorem8.1,
p.297).
Let 0 be a
G-orbit
inH.
SetN(O) = {03C1~|03C1|H
lives onO} Clearly (O)
isG-invariant.
LEMMA 5.3.2: The set
of G-orbits
inN
contained inN( 0)
is countable.PROOF: First we observe that any element in
N
lives on an N-orbit inH, since
H isregularly
imbedded in N(cf. 5.2).
Thenpick
7r e O.Clearly
each
G-orbit
in(O)
contains at least one element from(03C0).
Thelemma then follows from Lemma 5.3.1.
LEMMA 5.3.3: Each
G-orbit 0 in fI
carries a non-zero,positive,
inva-riant Radon measure.
PROOF: Let
R
be theanalytic subgroup
inG corresponding
to r. PickTceO. First we show that
03C0
is a closedsubgroup
inG:
Leta:
/03C0 ~
0 be themap s
~ s03C0. Then a is ahomeomorphism ([12]
Theorem
1,
p.124). Also,
theR-orbit ·03C0
c O islocally
closed(loc. cit.),
hence
locally compact
in thetopology
induced from H. But then also03C0/03C0
islocally compact
in thetopology
induced from/03C0,
hence03C0
islocally compact ([17] Theorem,
p.52).
But thenRGn
is a closedsubgroup.
We next observe that since[R, R]
c H c03C0
we have thatRGnlGn
carries the structure of an abelianlocally compact
group, hence it carries a non-zero,positive,
invariant Radon measure v. Let then S be theanalytic subgroup corresponding
to the Levisubalgebra
5. S is acompact
group. Define the map 8 : S x/03C0 ~ GIGn by
03B5 :(s, i)
-(s, r) :
This map is
clearly welldefined, surjective
and continuous. Moreover it is proper. Infact,
if C is acompact
subset ofthen 03B5-1(C)
is closedand contained in S x
(SC
n03C0/03C0),
which iscompact
since03C0/03C0
isclosed in
/03C0 (cf. above).
It follows that03B5-1(C)
iscompact.
We have thusproved
that B is proper. Define then the measure p to be theimage
of
ds (S)
v, where ds is a Haar measure on S.This y
iseasily
seen to be anon-zero,
positive,
invariant Radon measure on/03C0. This
proves the lemma.LEMMA 5.3.4: Each
G-orbit
in carries a non-zero,positive,
invariant Radon measure.PROOF: This is
clear,
since[G, G]
c N cG p
for all03C1~,
and therefore/03C1
carries the structure of an abelianlocally compact
group.For a
G-orbit
0 infi,
ket vo be the(essentially unique)
non-zero,positive
invariant Radon measure on 0(Lemma 5.3.3.).
We define theequivalence
class ofrepresentations II o
of Hby
(cf. [9]
8.6.3.Définition). Similarly,
if (9 is aG-orbit
in we let vo denote the(essentially unique)
non-zero,positive,
invariant Radon measure on(9
(Lemma 5.3.4).
We define theequivalence
class ofrepresentations Po of N by
(cf. loc. cit.).
LEMMA 5.3.5:
indH ~N03A0O
isquasi-equivalent
to~~ (O)P.
PROOF: Set X =
GIG1t,
y =/03C0N
and Z =G1tNIG1t’ observing
thatG1tN
is a closedsubgroup
in G. Let dx be the invariant measure vo on X and letdy
be an invariant Radon measure on Y(existing
since Y carriesthe structure of an abelian
locally compact group).
But then also Z carries an invariant measuredz,
say, which can be normalized such that(cf. [2],
p.95).
Let a:
GIG1tN
= Y -G
be a Borel section with03C3(03C0N)
= e and let03C31:
G1tNIG1t
= Z -G1tN
be a Borel section with03C31(03C0)
= e and suchthat
03C31(z)~N
for all zeZ(this
ispossible
sinceG1tNIG 1’-1 N/N03C0).
Nowthere is defined a Borel
isomorphism
Y x Z - Xby (y, z)
~c(y)z.
Wecan
extend ul
toGIG1t
= X as a Borelsection,
also called 03C31, such that03C31(x) = 03C3(y)03C31(z)
if x =03C3(y)z.
Having
thissetup
we can describelI o by
on the Hilbert space
sr Hx ds,
where x ~Hx
is the constant field ofHilbert spaces with
Hx = H03C0 (here
weidentify /03C0
with0).
ThereforeindH ~N03A0O
can be describedby
the directintegral
on the Hilbert space
sr xdx ~ L2(X, dx, H),
where =indHrN1t
andwhere x ~
Hx
is the constant field of Hilbert spaces withflx
=Hn.
Define the
unitary
R onL2(X, dx, H) by RF(x) = (03C31(z))F(x),
wherex = c(Y)z.
ThenRO(s)R-1F(x)=(03C31(z))(03C31(x))(s)(03C31(z)-1)F(x) = (03C01(z)03C31(x)-1s03C31(x)03C31(z)-1)F(x) = (03C3(y))(s)F(x), where x = c(y)z. Now identifying L2(X, dx, H)
withL2(Y, dy, H) ~ L2(Z, dz, )
via the above Borelisomorphism
we see thatO
isquasi-equivalent
to therepresentation
Let then
03C1 ~ (03C0).
We haveG N ~ 03C1
andG N/Gp
is abelian. LetX = /03C1
andZ = G1tNIGp
andlet 1:Z ~ G1tIJ
be a Borel section with03C31(03C0) = e.
As above we define a Borelisomorphism
Y x
Z ~ X : (y, z) ~ 03C3(y)z
and extendalto X
so that03C31(x)
=u(y)a 1 (z)
ifx =
6(y)z. Also,
there are invariant measuresdi, dz,
such that dx onX corresponds
tody Q
dz on Y xZ
via the aboveisomorphism.
Let us then observe that there exists a Borel function z ~
U.
fromZ
into the
unitary
group ofH
such thatUz(s)Uz-1 = 03C31(z)(s).
To seethis we first note that since
G 1t
stabilizes the irreduciblerepresentation
03C0of H there exists a Borel function t ~
v(t)
fromG1t
into theunitary
group ofH03C0,
such thatv(t)03C0(s)v(t)-1
=(t03C0)(s).
But then there exists a Borel function t ~V(t)
fromG1t
into theunitary
group ofH,
such thatV(t)(s)V(t)-1
=(tii)(s).
Next we find Borel functionsa: G1tN --+ G1t
and b :
G1tN --+ N,
such that s =a(s)b(s),
s ~G1tN.
The existence of such functions a and b areeasily
verifiedusing
the existence of Borel sections on coset spaces of closedsubgroups.
Define thenfinally Uz
=(b(03C31(z)))V(a(03C31(z))).
Then z- -U.
is a Borel function and it iseasily
seen thatUz(s)Uz-1 = 03C31(z)(s)
for z EZ,
s E N.Now
arguing
as above we find thatis
quasi-equivalent
toand
comparing
this to what we saw above we find thatflo
isquasi- equivalent
toWe then
apply
the above to a fixed element po E9(n)
and recall that 7r isquasi-equivalent
to~03C1~(03C0)03C1 (Lemma 5.3.1),
from whichflo
isquasi- equivalent
toLetting o
be the G-orbitthrough
po we have thatP,.
isrepresented bv
The conclusion of all this is then that
O
isquasi-equivalent
to arepresentation,
which containsPmo
as asubrepresentation,
and this is sofor all
G-orbits Wo
contained inN(O).
Nowtaking
amultiple
of110
wethen arrive at the
following preliminary
result:O is quasi-equivalent to
a
representation
which contains~ ~ (O)P
as asubrepresentation.
We then consider the other direction of the lemma. As we saw
170
isquasi-equivalent
towhich, by
Lemma5.3.3,
isquasi-equivalent
toFor a
given C-orbit (9
c(O)
and p ~ nN(71:)
we also saw, with the notation fromabove,
thatP,
is describedby
Let us then observe that
Z
is countable. Infact, GpN
acts as a trans-formation group in
9(n),
which is countable(lemma 5.3.1),
and thereforeGpNIG1t
=Z
is countable. It follows that we can write the represen- tation(*)
asThe summand
corresponding
to z =Gp
is~~Y 03C3( y)03C1 dy
from which weconclude that
~~y 03C3(y)03C1dy
is asubrepresentation
ofPw.
But then~03C1~(03C0) 10 y u(y)p dy
isequivalent
to asubrepresentation
of~03C1~(03C0)PO(03C1),
where
(9(p)
is theG-orbit through
p. Nowclearly ~03C1~(03C0)PO(03C1)
isquasi- equivalent
to~~(O)P.
We thus arrive at thefollowing
result:~~ N(O)P()
isquasi-equivalent
to arepresentation
which has asubrepre-
sentation
quasiequivalent
toiÎo. Comparing
this with ourprevious
resultwe
finally
arrive at the conclusion that~~(O)P
isquasi-equivalent
tofi,.
This proves the lemma.Assume now
that x : ~ Rt
is a continuoushomomorphism
withkerx -:)
N and that po is a non-zero,positive, x-relatively
invariantRadon measure on O. Then also each
6-orbit (9
inlV(O)
carries a non-zero,
positive, relatively
invariant Radon measure withmultiplier
x, ,uo, say. Letfo
be the(G, x)-character
on Hgiven by
(cf. [21] Proposition 5.1.4)
and letfo
be the(G, x)-character
on Ngiven by
(cf.
loc.cit.)
LEMMA 5.3.6: The