C OMPOSITIO M ATHEMATICA
I. M. G ELFAND M. I. G RAEV A. M. V ER ˇ TIK
Representations of the group of smooth mappings of a manifold X into a compact Lie group
Compositio Mathematica, tome 35, n
o3 (1977), p. 299-334
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299
REPRESENTATIONS OF THE GROUP OF SMOOTH MAPPINGS OF A MANIFOLD X INTO A
COMPACT LIE GROUP
I.M.
Gelfand,
M.I.Graev,
A.M. Ver0161ikAbstract
Some
important
nonlocalrepresentations
of the groupGx
con-sisting
ofC°°-mappings
of a Riemannian manifold X to a compactsemisimple
Lie group G are constructed. Theirreducibility,
as well asnon-equivalence
of the introducedrepresentations corresponding
todifferent Riemannian metrics are
proved.
Thering
ofrepresentations
is calculated.
Noordhoff International Publishing
Printed in the Netherlands
Introduction
In this paper the
unitary representations
of the groupGx
of smooth functions on a manifold Xtaking
values in a compact Lie group Gare
being
built.According
to the idea of a paper[20],
one can construct irreducible nonlocalrepresentations
of the group of measurable G-valued func- tions onX,
Gbeing
a Lie group, in the case when theunity representation
is not an isolatedpoint
in the space of irreducibleunitary representations.
Such a construction for groups G =SU(n, 1)
and G =SO(n, 1)
is realized in[20]
and[21].
The otherexamples
ofgroups of this kind
represent
the groups of isometries of a Eucledean space, as well as solvable andnilpotent
groups(cf.
Araki[2],
Streater[17],
Guichardet[8], [9],
Delorme[5], Parthasarathy-Schmidt [13]
etal.).
In the case of
compact
Lie groups G theunity representation
is anisolated
point,
and it isimpossible
to follow the scheme of[20]. If,
however,
one takes the groupGx
ofcontinuously
differentiablemappings
instead group of measurablemappings
then it ispossible
toconstruct for fhis group a series of irreducible nonlocal represen- tations. The idea of this construction is to consider at first the group
0’(X; G)
of smooth sections of the1-jet
fibrebundle j 1(X ; G) ---> X (cf.
§1);
the initial groupGx
isnaturally
imbedded in the0’(X; G)
as asubgroup.
As03B81(X ; G)
may beregarded
as a group of functionstaking
values in theskew-product
G .(# x ... x #),
#being
the LiedimX
algebra
of the group G(cf. §I),
and as theunity representation
of thisskew-product
is notisolated,
so it ispossible, following
thepattern
of[20],
to build nonlocal irreduciblerepresentation
for03B81(X ; G)
and to restrict it on thesubgroup GX.
Here in this paper westudy
therepresentations
of the groupGx
obtained in this way.The constructions of this kind of the
representations
of the groupGx
have beenfound,
after the appearance of[20], by
the authors of thepresent
paperand, independently, by Parthasarathy
and Schmidt[14],
R.S.Ismagilov [11],
Albeverio andHoeg-Krohn [1].
The ir-reducibility
of theserepresentations
for G =SU(2)
in the casedim X > 5 had been
proved by
R.S.Ismagilov [11].
As communicated to the authors A.
Guichardet,
P. Delorme had been studied therepresentations
of the groupGx,
where G is acompact
Lie group. It isbeing proved
in the present paper theirreducibility of
therepresentations for
the groupGx,
where G is any compactsemisimple
Lie group and dim X > 2.The case dim X = 1 remains open at the moment, the
difficulty being
connected with the morecomplicated,
than for thespaces X
ofgreater dimension,
character of the restriction of therepresentation
ofthe group
Gx
on thesubgroup AX,
where A C G is the Cartansubgroup.
We use in this paper, in a
systematic
way, thetechnique
ofGaussian measures, which has been for the first time used in the
study
of nonlocal irreduciblerepresentations
of current groups in[20], [21] ]
and which hasproved
very useful forverifying
the ir-reducibility
andnon-equivalence
of functional grouprepresentations (see
also[22]).
Here is the brief contents of the paper.
We
consider, in § 1, jet
fibrations over X and the groups connected with them and weexplain
the main idea of theconstruction
of therepresentation.
We also introduce there the Maurer-Cartancocycle,
whichplays
asignificant
role in this construction. The§2
is ofsubsidiary
character. There we
give
an account of what is connected with theconstruction
EXPO
T. This construction is more or lessexplicitly
described in
[2], [8], [13],
but the usage of Gaussian measures, whichbegan
in[20], [21],
allows todevelop
asystematic theory.
It will be set forth in detail somewhere else. In§3
we construct therepresentations
ofthe group
Gx
and formulate theprincipal
results of the paper. Theproofs
of the main theorems aregiven
in§§4
and 5.In some papers of
physical
nature(see,
forinstance, [ 18], [3])
it was considered the so-calledSugawara algebra.
The authors have noticed that thecorresponding
group is the centralR1-extension
of the group03B81(X ; G) (see above). (If
we take G =R 1,
then we get thegeneralized
functional
Heisenberg group).
We construct, in§6,
nonlocal ir- reducibleunitary representations
for this group as well.§ 1.
Thejets
of smooth functions with values in a Lie group and Maurer-C artancocycle
Here we introduce the
principal
definitionsconcerning
the groupC~(X ; G):
the fibre bundle ofk-jets,
the group of sections of this fibrebundle, k-jet imbeddings
etc. Thedescription
of the mostimportant
classes of
representations
of the groupC~(X ; G),
both local andnonlocal,
becomes moretransparent
if one passes to the group of sections of thek-jet
fibre bundle(cf.
section 3 of thisparagraph).
Themost
important
for our purposes is the definition of the Maurer- Cartancocycle, given
in section 2.1. The group
GX - C~(X ; G) and its jet
extentions. Let G be anarbitrary
real Lie group, X - a real connectedC°°-manifold.
Consider the setC~(X ; G)
ofC°°-mappings g :
X --> G such thatg(x)
= 1 outsideof some
compact
set(depending
ofg).
Let ussupply
the setC~(X ; G)
with the naturaltopology.
The groupoperation
inC~(X ; G)
is definedpointwise: (9192)(X)
=gl(X)g2(X).
We shall denote thetopological
groupC-(X; G) by Gx.
Let us define now the
k-jet imbedding
of the groupGx (k
=0, l, ...).
Recall that thek-jet
of amapping
X - G in apoint xo E X is, by definition,
the class of smoothmappings
X -->G,
all of themtaking
the same value at xo, and such that allcorresponding partial
derivatives of these
mappings
up to the k-thorder,
taken at xo,coincide. One can define the
k-jet
space at apoint xo E X
as follows.Consider a
subgroup GXXO,K
ofGX, consisting
of such functionsg: X -->
G that
g(xo)
= 1 and allpartial
derivatives ofg
up to the k-th order areequal
to zero in apoint
xo. It is easy toverify
thatGXXo,k
is a normalsubgroup
of the groupGX ;
the space ofk-jets
in apoint
xo isnaturally
identified with the factor groupGX/G ô,k.
It is clear that the
factor-groups GX/G k corresponding
to differentx E X are
isomorphic.
The groupGxlGxx is,
moreover,uniquely defined,
up toisomorphism, by
the groupG,
number k and dimensionm of the manifold X. We shall denote this group
by Gm
and call it the Leibnitz groupof
order k anddegree
mof
the group G.In what follows we consider for the most part the case k = 1. Then
G’ - G - (&
x... X &),
with & - the Liealgebra
ofG,
i.e.Gm
is am
skew-product
of the direct sum of mcopies
of Liealgebra
OE and thegroup
G; representation
of the group G on OE x ... x (S isadjoint representation.
The groupGi
caneasily
be described as well(see [14]).
In thegeneral
case the groupGm
is askew-product
of G and anilpotent
groupUm,
which space is a sum of severalcopies
of thespace
OE ;
the formula of the group rule inGm
forarbitrary k
and m isactually
rathercomplicated.
Let us define a
k-jet
fibre bundlejk(X; G) > X,
which is a fibrationover X with fibre
corresponding
to x E Xconsisting
of all thek-jets
in a
point
x. The structure of a fibre bundle is introducedin jk(X ; G)
in a naturalway.’
The fibre bundle
jk(X; G) gives
reason to thefollowing
definitiongeneralizing
the definition of a vector bundle:DEFINITION: Let H be a connected Lie group. A smooth fibre
bundle e
with the standard fibre H and the structure group Aut H(=
the group of all continuous groupautomorphisms
ofH)
will becalled a group bundle with a group H.
A group bundle with an additive group Rn
(or Cn)
is a vector bundlein usual sense, that is
why
our notionpresents
a ’noncommutative’analogue
of a vector bundle. The sections of a group bundle form agroup, since each fibre has a group structure. A trivial group bundle with a group H and base X is a
fibering
H x X -->X,
and the group of all its smooth sections is the groupHx.
The
jet
fibrebundle j k (X ; G) ---> X
defined above is a group bundle in the sense of our definition with the groupGm,
m = dim X. Thisbundle is not trivial unless the
tangent
bundle be so.Every
fibre of this bundle over x E X iscanonically provided
with a structure of thegroup
G xl Gx GM.
Let us denote
by 03B8 k(X ; G)
a space of all differentiablesections,
’It is a more traditional approach to consider
jk(X ;
G) to be a fibre bundle over X x G. The definition in the text is however, more suitable for us.with
compact
support, ofk-jet
bundlejk(X ; G) ---> X supplied
withusual
topology.
The space03B8 k (X ; G)
is a group withregard
topoint-
wise
multiplication
in the fibres.Notice that there are defined natural group
epimorphisms
0k+I(X ; G)- ok(X ; G), k = 0, 1,... ;
inparticular, epimorphism 0’(X; G) --> 0°(X; G)
=GX.
DEFINITION: We call a
k-jet imbedding
a mapdefined
by: (Jkg)(x)
is thek-jet
of a functiong
in apoint
x E X.The
following
isfairly
evident:Jk is a
groupmonomorphism.
REMARK: All
proposition
of this section remains true if one sub- stitutes the groupGx by
the group of differentiable sections of any group bundle over X with the groupG, jk(X ; G) by
thecorresponding k-jet
fibre bundle etc.2. The group
03B81(X ; G)
and Maurer-Cartancocycle.
As we shall deal with the case k = 1 let us examine the group(J1(X; G)
in greater detail. For every x E X the elements of a factor groupGXIG;l
may be considered as thepairs (g(x), a (x)),
whereg(x) E G and a (x)
is a linearmapping
oftangential
spacesTXX-Tgx>G,
i.e.a (x) E Hom (Tx, Tg(x»). Consequently,
one canput
intocorrespondence
toeach element of the group
03B81(X ; G)
a map TX --> TG of thetangential
sheaves which is linear on the fibres. It is easy to see that this
correspondence
is anisomorphism
of the group01(X; G)
and thegroup
(TG)TX
of all differentiablemappings
TX - TG withcompact support,
which are linear on the fibres.Further,
theright
trivialization of thetangential
shief TGpermits
us to
identify
eachtangential
spaceTgG
with the spaceTeG ~
# and thus defines anisomorphism
TG G - # of the group TG and a semidirectproduct
G - #. Therefore the group(TG)TX
isisomorphic
to a
skew-product
of the groupGx
and an additive group03A91(X ; #)
ofall differentiable
mappings
’TX->@ withcompact support
and linearon the
fibres,
i.e. a group of (S-valued 1-forms on X.Consequently,
there are determined the canonicalisomorphisms
ofthe groups :
where
03A91(X; #)
is the space of all differentiable #-valued 1-formswith
compact support
onX,
and the action V ofGx
on03A91(X; OE)
isan
adjoint
action in every fibre:REMARK: We have two different
imbedding
ofGx
into03B81(X; G):
J1 : GX ~ 03B8’(X ; G)
andG x --> G x - 0 C G x - f2’(X; (9) -
Since
Gx
isacting
onf2’(X; (b)
and there is a naturalepimorphism 03B81(X; G)- GX,
then it is defined in03A91(X;#)
also an action of thegroup
03B81(X; G).
Let us introduce the Maurer-Cartan
1-cocycle. Making
use of theisomorphism 03B81(X; G) ~ Gx - f2’(X; W),
define a1-cocycle
a of03B81(X ; G) taking
values inf2’(X; (M) by
thefollowing
formula. Givene
E8’ (X ; G), 1.e. i = (g, (o), where g E Gxl .
En’(X; @),
letDEFINITION: A restriction of the
1-cocycle a
on thesubgroup
J1GX ~ G’
will be called Maurer-Cartancocycle
and denotedby 03B2.
Therefore
For
03B2
to be a1-cocycle
means that for everygh g2 E GX
it satisfiesIt is not hard to prove that the
conception
of the Maurer-Cartancocycle
is natural withrespect
to thediffeomorphisms
of manifoldsX --> Y.
Owing
to the extremeimportance
of the Maurer-Cartancocycle
weshall
give
a direct definition of it. Denoteby dg
a differential of a mapg : X - G,
Let R : TG - OE be the
right
trivialization. Then we haveREMARK 1: Let : TX - TX be a map,
preserving
each fibre and linear on it. Then the mapis a
1-cocycle
to the groupGx again.
The set of suchcocycles
forms a linear space. It isinteresting
to find out whether every1-cocycle
ofGx
with values inf2l(X; @)
iscohomological
to acocycle
of the form(2) with 8
the Maurer-Cartancocycle.
REMARK 2: The group
G1m ~
G -(@
x ... x@)
isrepresented
withm
invariant
scalar-product
in the space @ x ... x (S and possesses am
1-cocycle
with values in this space, definedby
The
1-cocycle a
of03B81(X; G)
introduced above can be definedlocally
in the
following
way. Assume that we have some local coordinates inX,
and so theisomorphism
Hom(Tx, @) ~
@ x ... x@, m
= dim X ism
defined. Thus the formula for the
cocycle a
may be written asfollows:
Thus the existence of the Maurer-Cartan
cocycle
for a groupGx
is connected with the existence of the1-cocycle
ao forGm.
3.
Representations of
the groupGx. By using k-jet imbeddings
Jk : Gx ~ 03B8k(X ; G)
one can construct variousrepresentations
ofGx.
Consider at first
arbitrary unitary representation
ir of the groupGm.
Fix a
point xo e X
and define anisomorphism GX/G ô,k ~ Gm.
Arepresentation
of the groupGx
can be definedby
The
representation (3)
is ananalogue
of thepartial
derivative in xo.For
instance,
letG =R1,
thenG=R1(f)Rm;
if 03C0 is a one-dimen-sional
representation
ofGm : (03BEo, 03BE1, ... , 03BEm)~exp (i m o as03BEs),
thenT’1T(g)
is a one-dimensionalrepresentation g-exp[1(aog(x)+
3lJJf=1 Qs(aglaxS))x=XQ].
For G =SU(2), k
= 1 and m = 1 such represen-tations were described for the first time in
[6]; they
were constructedthere
directly by passing
to alimit,
like that used in the definition of aderivative.
The
representations (3)
are local ones sincethey depend
onk-jets
in a
point xo E X only. Similarly,
one can definerepresentation
ofGX depending
onk-jets
in a finite number ofpoints.
More
interesting representations
of the groupGx
arise due to the existence of nonlocal irreduciblerepresentations
of the group03B8k(X ; G),
the restrictions of which toJkGX
remain irreducible.In
[20]
there wasgiven
a construction of nonlocalrepresentations
for a current group, i.e. a group of functions on X
taking
values insome Lie group. For that construction to be
applicable
is necessary that theunity representation
of the coefficient group be not isolated in the space of all itsunitary representations,
or, in moregeneral terms,’
that the first
cohomology
group with values in the space of someunitary representation
of the coefficient group be non-trivial.Since the local structure of the group
03B8k(X ; G)
is similar to that ofthe group
c~(X: Gm),
m = dimX,
the construction of[20], [21] can
beapplied
to0’(X; G)
if theunity representation
ofGm
is not isolated.That is the case when G is a compact
semisimple
Lie group and k =1,
forGm
then possesses anorthogonal representation
in the space@ x ... x OE
and,
as we have seen, there exists a non-trivial1-cocycle
m
of the group
Gm
with values in that space.For the group
Gx,
with G a compactsemisimple
Lie group, therearises, consequently,
a nonlocalunitary representation.
We em-phasize
thatthough
thisrepresentation
is notlocal,
thusresembling
the
representations
constructed in[20],
it does not admit any ex- tention even to the group of continuousmappings
XG,
because its construction makes use of the1-jet imbedding.
§2.
Some subsidiariespreliminary
factsIn the
present paragraph
we set forth somegeneral
definitions and statements, which will be used for the construction ofrepresentations
for
Gx,
as well as in theproofs
of the main theorems.1. The space EXP
H.
Let H be a real nuclearcountably
Hilbert-’ In all examples known to the authors the existence of a unitary representation of a
Lie group with non-trivial first cohomology group accompanies the non-isolation of the
unitary representation in the space of unitary representations. Moreover, such a representation admits a deformation which is at the same time a deformation of the
unity representation. However, the necessity of such coincidence is not yet established.
norm space, and let
(, )
be some innerproduct
in it. Denoteby H
acompletion
of H in the normllhll
=(h, h)1/2
andby
H’ a dual to H.Then there are natural
embeddings
HC H
C H’.Let us define in H’ a Gaussian measure u with the zero mean and a
correlation functional
B(hi, h2) = (hl, h2), by
its Fourier transformThis measure g will be called a standard Gaussian measure in H’.
Let us introduce a
complex
Hilbert spaceL 2(H’)
of all square-integrable
withrespect to g
functionals on H’. The functional f2 EL 2(H’), identically equal
tounity
onH’,
will be called the vacuumvector.
We shall determine a natural
isomorphism
of the spaceL 2H’)
andanother space which we call an
exponential
of H’ and denote EXPH.
Let
He
be acomplexification of H
andSnHe (n = 1, 2,...)
thesymmetrized
tensorproduct
of ncopies
ofHc.
Let alsoS°Hc
= C.Call an
exponential
EXPH of
a spaceH,
or Fock spacecorrespond-
ing to H,
acomplex
Hilbert spaceTo establish an
isomorphism
EXPfi == L,(H’)
consider in thespace
EXP fi
the set of vectorsThe set is known to be total in EXP
H [81,
i.e. its linear span is dense in EXPH.
Consider amapping
of the setlexp h}
into the spaceOne proves that this
mapping
conserves the innerproduct
and so,by
virtue of thetotality
of the sets{exp h}
and{ei(o,h)}
inEXP H
andL 2(H’) correspondingly,
themapping (1)
can beuniquely prolonged
up to an
isomorphism
of the spaces:We call thus defined
isomorphism
the canonicalisomorphism
betweenthe space
L,(H’) 2
and its Fock modelEXP H
=0 n=o Sn Hc.
Noticethat the canonical
isomorphism
puts incorrespondence
the vacuum vector f2 eL2(H ’)
and the vector 1 = exp0,
as well as thesubspace
of deneralized Hermitian
polynomials
ofdegree
n and thesubspace SnHc
inEXP H (see,
forinstance, [23], [12]).
REMARK: If there are two
triples
of spacesHIC 17, CH,,
H2 CH2 C
H2 withH-
=H2,
and /LI, IL2 are standard Gaussian measures in H Í and H2correspondingly,
then there exists a canonical isomor-phism L2{HD ~ L 2 2(H’)-
It arises from canonicalisomorphisms
between this space and the Fock space.
2. The
representation EXPI3
V. Consider atopological
group G andan
orthogonal representation
V of G in the space H. Let us extend therepresentation
V to the space H’ D H dual toH, by
for any
F E H’, h E H.
Consider a
l-cocycle 13
of the group G with values inH,
i.e. a continuousmapping 03B2 : G ~ H
which for every gt, g2 E G satisfiesGiven a
representation
V and a1-cocycle /3,
we shall construct a newrepresentation, U,
of the group G in the spaceL’(H’)
EXPH, by
Call the
representation
U anexponential
of the initialrepresentation
V of G
(with respect
to the1-cocycle /3)
and denote itEXPO
V. Nowlet us
point
out somesimple properties
of therepresentations EXP 13
V.(1)
Let the1-cocycle 03B2, 03B2: G ---> H
becohomological,
i.e. there exists such a vectorh0 E H
thatB’g -,Bg
=V(g)ho - ho
for eachg E
G. In this case therepresentations
U =EXPO
V and U’ =EXP 03B2’
V areequivalent.
We can
observe, indeed,
that for every g EG, U’(g)
=A% U(g)A-’
where
Ah,,
is definedby (A%O)(F)
=ei(F,hO>cp(F).
(3)
LEMMA 1. LetVc
be thecomplexification of
arepresentation V, snvc (n
=1, 2, ...)
asymmetrized
tensorproduct of
ncopies of
therepresentation Vc, SoVc
aunity representation. If 6
=0,
thenTo prove the lemma it suffices to pass to the Fock model of the
representation
space.Notice now that
given
al-cocycle 13 :
G - H and anarbitrary
bounded linearoperator
A in H which commutes with the operators of therepresentation V,
the functionis a
1-cocycle
of Gtaking
values inH,
too.Thus,
with every1-cocycle Q
one can connect a
family
ofunitary representations
of G inL 2(H,):
A
being
any bounded linearoperator
inH, commuting
with theoperators
of therepresentation
V.LEMMA 2
(of
a tensorproduct):
Let a =(a aIl aal2)
a21,a22 be anarbitrary
matrix
of
bounded linear operators aij: H -H, commuting
with the operatorsof
arepresentation
V andsatisfying
the condition :aa1j
+af¡a2j
=3ijE (i, j
=1, 2),
Ebeing
theunity
operator(i.e.
a* a =1).
Then: Define an
operator
inThe
operator R
isunitary since,
as it is easy to see,For every
whereof the statement of the lemma follows.
COROLLARY:
If Ai,
A2 are invertible operators inH,
thenwhere A =
(Ar Al
+Ar A2)1/2, Uo-the representation,
whichcorresponds
to zero
cocycle.
Indeed, put,
forshort,
Bi =A*A 1,
B2 =A2 A2
and notice that B1-B1A-2B1= B2 - B2A-2B2,
this operatorbeing self-adjoint
andposi- tively
definite. Consideroperators
a 11= A -1 Ar, a 12 = A -1 Ar,
One
easily
proves that theseoperators satisfy
the conditions of Lemma 2. On the otherhand, auA1 + a12A2 = A,
a21A1 + a22A2 =0, q.e.d.
3. The
definition of
therepresentation for
a group G with respect to apair of 1-cocycles.
Let U =EXPI3
V be therepresentation
of G in the spaceL 2(H’)
defined in the section 2. It is not difficult toverify
thatthis
representation
isequivalent
to thefollowing representation Ux
in the spaceL 2(H’):
A
being
anycomplex number, [A[ = 1. Namely, UA(g) = AAU(g)AÀB
where AA is the operator,
uniquely
definedby
its action on thefunctionals, ei(;h)
Given another
1-cocycle (3’
of the group G with values in H wedefine the
operators Ûx(g)
inL’(H’) by
THEOREM: The operators
ÚÀ(g)
areunitary
and constitute aprojective representation
of the groupG, namely
The
proof
is immediate.We want to define now an extension of the additive group
R+ by
thegroup G. For this purpose observe that the function a for every 91, g2, g3 E G satisfies
Consequently, a
is a2-cocycle
of the group G with values in R and therefore defines an extension6
ofR + by
G. The elements ofÔ
arethe
pairs (g, c),
g EG,
c ER +
with thefollowing multiplication
rule:It
corresponds
to theprojective representation
of G in the spaceL 2(H’)
defined above an(affine) unitary representation
of the groupG, given by
4. The
singularity
conditionsfor
two measures. Recall that two measures, liand v,
in the space X are said to beequivalent
if forevery measurable set A C X the conditions
IL(A) = 0
andv(A) = 0
hold
simultaneously.
The measures 1£ and v are calledmutually singular
if there is a measurable set A C X such thatIL(A)
=0, v(X - A) = O.
The lemma which follows is well known in the
theory
of the Gaussian measure spaces.LEMMA 3:
Let IL be the
standard Gaussian measure in the spaceH’. The measures p, and
li(- + x)
aremutually singular if
andonly if x ~ H.
PROOF: We may assume that
H = 12 and li
is aproduct
measure,IL = m1 x ... x mn X ... where
dmi(t) = (203C0)-1/2 e-11/2 dt;
x =(xl, ...,
Xn,
...).
The measureg
+x)
is aproduct
measure either and there-fore, by
virtue of the zero-onelaw,
the measures IL andIL(. + x)
are either
mutually singular
orequivalent. By
virtue of Kakutanitheorem the measures 1£ and
IL (. + x)
areequivalent
if andonly
ifIlk f Vdmk . dmk(.
+x)
> 0. The easy verification shows that thishappens
if andonly
ifk x 00,
i.e. xE H.
LEMMA 4:
Let IL
be the standard Gaussian measure in H’ and v a measure in H’satisfying v(H)
= 0. Then the measure IL ismutually singular
with the convolution IL * vof
the measures IL and v.(A
convolution of the measures IL and v is definedby (IL
*v)(.) = f IL(. - x)dv(x).)
PROOF: Assume that the measures IL and 1£1 = g * P are not mu-
tually singular.
Then(di£lldi£)
= p > 0 on a set ofpositive
03BC-measure.Since
IL1(.) = f IL(’ - x)dv(x) then, by
the Fubinitheorem, p (y)=
f [di£ (y - x)ldIL(y)]dv(x)
> 0 for y in the set mentioned above.Hence,
there exists a setB, v(B) > 0,
such that[d>(. - x)ldg]
> 0 for each x E B. Butthen g
and1£ (- - x)
are notmutually singular,
and con-sequently, by
Lemma3,
xE H,
in which casev(H)
> 0. We came to acontradiction with the
assumption.
5. The
spectral
measures. Let G be an abelian(not necessarily
lo-cally compact) topological
grouppossessing
a countable base of open sets,G
the group of its continuouscharacters,
U aunitary
represen- tation of G in thecomplex
Hilbertspace H. By applying
thespectral
theorem
(see,
forexample, [4])
to theC*-algebra, generated by
theoperators
of therepresentation U,
we get thefollowing
decom-position :
There exists anisomorphism of
thespace X
onto the directintegral of
Hilbert spaces,with 03BC a Borel measure on
G,
whichtransfers
the operatorsU(g),
g E G into the operators
The measure g on
G
is definedby
Uuniquely
up toequivalence
and is called the
spectral
measure of therepresentation (3).
Therepresentation (3)
of G in the spacefG Hxd>(x)
is called thespectral
decomposition
of the initialrepresentation
U. If dimHx
= 1 for a.e. X it is said that therepresentation
U has asimple
spectre.We
give
here some statements of thespectral
measures.(1)
Tworepresentations
of G aredisjoint
if andonly
if theirspectral
measures aremutually singular.
(2)
Thespectral
measure of the sum of tworepresentations
of G isequivalent
to the sum of theirspectral
measures.(3)
Thespectral
measure of the tensorproduct
of two represen- tations of G isequivalent
to the convolution of theirspectral
measures.
(4)
Theweakly
closedalgebra generated by
theoperators (3)
in the spacef IP Ydi£ (X)
coincides with thealgebra
of the operators ofmultiplication by
anarbitrary u-measurable
functiona (X): f (X) -a (X)f (X) (i.e.
thisalgebra
isisomorphic
toL,’ (C;),
see, forexample, [4]).
In the
§4
we shall make use of thefollowing generalization
of thestatement
4):
Assume that a
unitary representation
of the group G can bedecomposed
into a directintegral
ofrepresentations
(3
is a measure space with the measurev).
It means that U is
equivalent
to therepresentation
in the directintegral
of Hilbertspaces Ye = f: H03BEdv(03BE) given by
U03BE being
arepresentation
of G inH03BE.
LEMMA 5:
If
therepresentations U03BE1, U03BE2,
aredisjoint for
almostevery
(with
respect tov) 03BE1 ~ e2,
then theweakly
closed operatoralgebra generated by
the operatorsU(g),
g E G contains the operatorsof multiplication by
every bounded v-measurablefunction a (03BE) : f(03BE)~a(03BE)f(03BE).
COROLLARY: Let a
representation
Uof
G bedecomposed
in atensor
product of
tworepresentations,
U =U’o U’,
with the cor-responding spectral
measuresji’ andu". If
the measuresIL’(.
+Xi)
andIL’(’
+X2)
aremutually singular for
almost everyX1 ~
X2(with
respectto
IL"),
then theweakly
closed operatoralgebra generated by
the operatorsU(g),
g EG,
contains all operatorsE@ U"(g),
g E G(and,
therefore,
all operatorsU’(g) 0 E).
Indeed,
letU" = f! U"di£"(x)
be thespectral decomposition
of therepresentation
U". Then U =f G (U’@U()d>"(x).
Since thespectral
measure of the
representation U’o U,’
isIL’(.
+X), then, by
theconjecture
madeabove,
therepresentations U’ @ U’X,
andU’ @ U§g
are
disjoint
for almost all(with respect
toIL") Xl "#
X2. It follows from the Lemma 5 that theweakly
closed operatoralgebra generated by
the
operators U(g), g
EG,
contains theoperators
ofmultiplication by
the functions
ag(X)
=X(g),
i.e. theoperators E@ U"(g).
§3.
Nonlocalreprésentations
of theGroup G .
Thering
ofrepresentations
1. Construction
of
therepresentations of
the groupGx.
Let usbegin
to
study
of therepresentations
ofGx
which are connected with the Maurer-Cartancocycle (see § 1).
From now on we shall consideronly
those Lie groups G
for
which their Liealgebra
possesses an innerproduct
invariant under theadjoint
actionof
the group G. In par-ticular,
all compact and all abelian Lie groupssatisfy
this condition.In order to construct a
representation
ofGx
we assume that X hasa structure of a Riemannian manifold. This structure induces an
orthogonal
structure in the tangent bundleTX,
as well as astrictly positive
smooth measure dx on X.Let us introduce an inner
product
in the space03A91(X)
of R-valued 1-forms of the COOclass,
with compact supportby
the formulawhere
(,)x
is an innerproduct
in theconjugate tangential
spaceT-*X.
Let us also fix an inner
product
in the Liealgebra
K of G which is invariant under theadjoint
action of the group G.Consider now the space
il1(X; 0153) = il1(X) @ 0153
of differentiable K-valued 1-forms on X with compact support. Theorthogonal
struc-tures in the spaces
03A91(X)
and K introduced above induce the or-thogonal
structure in their tensorproduct f2’(X; K).
It is clear that the latter innerproduct
in il1(X ; K)
isG X-invariant.
Denote
by
H thepre-Hilbert
spacef2’(X; K), by H
itscompletion
in the norm introduced in
H, by fF
a spaceconjugate
to H andby IL
the standard Gaussian measure in 3W.
Let us
define, according
to thegeneral
definitiongiven
in thesection 2 of
§2,
the newunitary representation
U =EXPO
V of thegroup
Gx.
For that purpose we shall extend therepresentation
V ofG X
f rom H to thespace F D H by
the formulaLet
03B2
be a Maurer-Cartan1-cocycle,
that is03B2g = R -dg (see § 1 ).
Define a
unitary representation
U of the groupGX
in the spaceL2 m (F)
by
Note that the
representation
Udepends
on the Riemann space structure in X.We formulate now the main results of the paper.
THEOREM 1:
If
G is a compactsemisimple
Lie group and dim X >2,
then therepresentation
U =EXPG
Vof
the groupGx
is ir- reducible.THEOREM 2: Let G be a compact
semisimple
Lie group. Then therepresentations
Uof Gx corresponding
todifferent
Riemannianmetrics on X are not
equivalent.
The
proof
of the theorems1,
2 will begiven
in§5.’
It rests upon the results of§4
where the restriction of therepresentation
U ofGX
toan abelian
subgroup
is studied. Note that the main results of§4
are valid for the case dim X > 2only.
Theproblem
ofirreducibility
forthe
representation
U in the case dim X = 1 still remains open.REMARK 1: The
representation
U ofGX
is a restrictionto J1 Gx ~
GX
of therepresentation Û
of the group03B81(X; G) ~ Gx . f2’(X; K)
inthe space
L 2(F)
defined in thefollowing
way(see § 1 ).
Let aE(J1(X;O),
i.e. a =(g,w), gEOX, wEn1(X;).
ThenThe
representation !7
is an"integral
ofrepresentations"
in thesense of
[20, 21].
Itsirreducibility
can beeasily
derived from thetheorems of
[20, 21].
Thus theorem 1 isasserting
that the represen- tationÏ7
remains irreducible when restricted to theimage
ofGx.
’ The notion of a spherical function of the representation U given in §5 allows one to prove Theorem 2 in a différent way (independently of §4).