数理解析研究所講究録
1017 巻 1997 年 116-123

116

1-COCYCLES ON INFINITE -DIMENSIONAL SPACES
BY

HIROAKI SHIMOMURA(福井大学)

下村宏彰

1. INTRODUCTION
acts as a measurable) on which
Let us consider a a-finite measure space (X,is equivalenttransformation group. We assume that is G-quasi-invariant. That is,map
is the image measure of by the$x-gx$ .
, for all $g\in G$ , wheretoof is defined as follows,
It follows that a unitary representation$\mathfrak{B},$

$G$

$\mu$

$\mu_{g}$$\mu$$\mu(\mu_{g}\simeq\mu)$$\mu$$\mu_{g}$$G$$(R_{\theta}, \mathrm{L}^{2}(\mu x))$(1.1)

$R_{\theta}(g)$:$f(x)\in \mathrm{L}_{\mu}^{2}(X)-\theta(x,g)\sqrt{\frac{d\mu_{g}}{d\mu}}(x)f(g^{-}1_{X)}\in \mathrm{L}_{\mu}^{2}(X)$,, is a -valued function on $G\cross X$ such that
where , so called
is a measurable function of , and(1) for each fixed $g\in G,$$\theta(x,g_{1})\theta(g^{-1}1X,g2)=\theta(x,g_{1}g2)$.
$g_{1},g_{2}\inG,$for(2) for all
Moreover if a group topology is induced to and the following condition (3) is satisfied,
we say that is continuous.
(3) $\theta(x,g)arrow 1$ in , if $garrow e$ in .
A simple example of 1-cocycles is the one described below which is so called l-coboundary,$\theta$$S^{1}$$1-\mathrm{C}\mathrm{O}\mathrm{C}\mathrm{y}\mathrm{C}\mathrm{l}\mathrm{e}^{*}$$x$$\theta(x,g)$$\mu- \mathrm{a}.\mathrm{e}.X$$G$$\theta$$\tau$$\mu$$\theta(x,g)=\frac{\phi(g^{-1}x)}{\phi(x)}$,where is a -valued measurable function. In this report we will pick up infinite dimensional one linear space and two groups as and will discuss on the 1-cocycles, especially
its characterization, connecting with canonical representations defined by (1.1).
$\phi$$S^{1}$$G$The names, 1-cocycle and 1-coboundary, come from group cohomologlical theory
be awhich is privately comununicated by Y.Yamasaki. Let us explain it briefly. Let
letFurther
left.
the
acts
on
from
that
assumegroup, be an Abelian group andis the same copy of for all $1\leq i\leq m$ .
to , where
be a set of all maps fromsuch thattobe a map fromPut$(\mathrm{c}\mathrm{f}.[10])$

$(*)$

$G$

$G$

$A$

$\Pi_{i=}^{m_{1}}G_{i}$$\partial_{m}$$\mathcal{F}_{m}$

$A$

$A$

$\mathcal{F}_{m}$$G$$G_{i}$$\mathcal{F}_{m+1}$$.( \partial_{m}\varphi)(g1, \cdots,g_{m+1}):=\sum^{m}(-1i=0+1)^{i}\varphi i(g_{1}, \cdots,g_{m+1})$

,

where
$\varphi_{0}(g_{1}, \cdots,g_{m+1}):=g1\varphi(g2,g_{3}, \cdots,g_{m+1})$$\varphi_{1}(g_{1}, \cdots,gm+1):=\varphi(g1g2,g_{3}, \cdots,g_{m+1})$$\varphi_{i}(g_{1}, \cdots,g_{m+1}):=\varphi(g1,g2, \cdots,g_{i}gi+1, \cdots,g_{m+1})$$\varphi_{m+1}(g1, \cdots,g_{m+1}):=\varphi(g_{1},g2\cdot\ldots,g_{m})$

.

,

117

is
, and
cohomology group
Then we have$a\inA$$(\partial_{0^{a}})(g):=ga-a$$m\geq1$elementin. An
for alland, wheredefined for) is called $m$-cocycle, ( $m$-coboundary), respectively. Now let us apply, (inthe above general theory to our situation. That is, we take as the equivalence class of
all measurable -valued function to modulo , and define the action of on such that$(gf)(x):=f(g^{-1_{X}})$ for all $g\in G$ and for all $f\in A$ . Then it is easily checked that l-cocycle(1-coboundary) is just the same with 1-cocycle (1-coboundary) in the cohomological sense.
$\partial_{m+1}\circ\partial_{m}=0$$H^{m}(G, A):=\mathrm{k}\mathrm{e}\mathrm{r}\partial_{m}/{\rm Im}\partial_{m-1}$$m\mathrm{t}\mathrm{h}$$\mathcal{F}_{0}:=A$${\rm Im}\partial_{m-1}$$\mathrm{k}\mathrm{e}\mathrm{r}\partial_{m}$

$A$

$A$

$G$

$S^{1}$$\mu$

Acknowledgement. I wish my thanks to Prof. T.Hirai at Kyoto University for
introducing me the subject in section 4. I also thank to Prof. H.Omori at Science University of Tokyo for giving me many valuable informations on the topics in section 4. In
particular the proof of Theorem 4.1 owe to him so much.

2. 1-COCYCLES DERIVED FROM COMMUTATION RELATION IN QUANTUM MECHANICS
of an infinite dimenFirst we shall consider 1-cocycles on the algebraic dual spacesional real linear space $X$ , which come from the representation of commutation relation
as the basic space and take a linear subspacein quantum mechanics. So we consider$X’$ ofas a transformation group such that for any $x\in X$ there exists $x’\in X’$ such
. The action
a natural duality bracket for $X$ andthat $<x,$ $x’>\neq 0$ , where $<.,$$x^{a}\mapstox’+x^{a}$. Now let us consider unitary representations
is defined by
of $X’$ on$X’$$X$
respectively which satisfy,
of and(1) $U(x)$ is continuous on any finite dimensional subspace of $X$ ,(2)
is cyclic, and(3) $U(x)V(X)’=\exp(\sqrt{-1}<x, x’>)V(X’)U(x)$ for all $x\in X$ and for all $x’\in X’$ .
Then the following theorems hold which are already well known.
$X^{a}$$X^{a}$$G$$X^{a}$$X^{a}$$\cdot>\mathrm{i}\mathrm{s}$$X^{a}$$(U, \mathcal{H}),$ $(V,\mathcal{H})$’$U$

on the cylindrical a-algebra
Theorem 2.1. There exist some probability measure$X^{a}\cross X’$and (V, ) aresuch that the representationsonand 1-cocycle on
realized as follows,$\mathfrak{B}$$\mu$$X^{a}$$(U, \mathcal{H})$$\theta$(2.1)$U(x)$(2.2)$V(x’)$::$f(x^{a})\in \mathrm{L}_{\mu}^{2}(X^{a})-\exp(\sqrt{-1}<x,x^{a}>)f(X^{a})\in \mathrm{L}_{\mu}^{2}(X^{a})$$\mathcal{H}$,.

$f(x^{a})\in \mathrm{L}_{\mu}^{2}(X^{a})-\theta(x^{a}, X;)\sqrt{\frac{d\mu_{x’}}{d\mu}}(x^{a})f(Xa-X’)\in \mathrm{L}_{\mu}^{2}(X^{a})$Theorem 2.2. (1) For the pair of representations $(U_{i}, V_{i})(i=1,2)$ which are defined byare equivalent to(2.1) and (2.2),
if and onty if the corresponding andare l-cohomologtA8. $i.e$ ,andare equivalent as measures and the corresponding
$uch$
that
.
there exists some 1-coboundary S.
(2) In order that the representation $(U, V)$ is irreducible, it is necessary and sufficientthat is $X’$ -ergodic. $i.e.$ , $\mu(A)=0$ or 1, provided that $\mu(A\ominus(A-x’))=0$ for all$(U_{2}, V_{2})$$(U_{1}, V_{1})$$\mu_{1}$$\theta_{1}$$\theta_{2}$$\mu_{2}$$\phi$$\theta_{1}=\phi\cdot\theta_{2}$$\mu$$x’\in X’$.

From the above theorems, we see that the pair of representations $(U, V)$ is characterized by two factors, that is, measure and 1-cocycle. So we shall look them quickly.
In the finite dimensional case, the problem is so simple. Namely, every translationally quasi-invariant measure is equivalent to the Lebesgue measure and every l-cocycle
is a 1-coboundary. While in the infinite dimensional case the situation is quite complicated. First of all there exist quasi-invarianr measures much enough to nonclassify them.
Secondly, even if a quasi-invariant measure is specified, there exist also many l-cocycles

118

which aredifferent from each other. Therefore it seems to be meaningless to
try to classfy 1-cocycles for a general . However it seems to be meani.ngful and important
to consider 1-cocycles for Gaussian measures picked up among many quasi-invariant
measures. Here the Gaussian measure on the algebraic dualof a Hilbert space $H$ is$\mathfrak{B}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$$\mu$$\theta$$g$$H^{a}$$g$defined by,

.

(2.3)

$\int_{H^{a}}\exp(\sqrt{-1}<x, x^{a}>)g(dX^{a})=\exp(-\frac{1}{2}||x||_{H}2)$

It is well known that is-quasi-invariant, where$H$ . So the problem becomes asfollows.
$g$$H^{*}$$H^{*}$.

is the topological dual space of(P) What kinds of 1-cocycles onfor the Gaussian measure do there exist ? Especially, it is a matter worthy to be considered when is continuous with the$\theta$$H^{a}\cross H^{*}$$g$$\theta$norm topology on$H^{*}$.

The following theorem is a modest result along this line.
$\cdot$Theorem 2.3. For any$s\in \mathrm{R}$consider a $continuo’\iota Ls$ l-cocycle$\theta_{s}(X, x)a*:=(^{\frac{dg_{x}*}{dg}(x^{a}}))^{\sqrt{-1}}S$(1) Then the canonical representations(2.4)$R_{s}(X^{*})$:$(R_{S}, \mathrm{L}_{g}^{2}(H^{a}))$defined by$f(xa)\in \mathrm{L}_{g}^{2}(H^{a})-\theta s(X^{a},X*)\sqrt{\frac{dg_{x}*}{dg}}(x^{a})f(x^{a}-X)*\in \mathrm{L}_{g}^{2}(H^{a})$give mutually inequivalent representations for all different .
(2) Letbe the image measure of by a homothety, $x^{a}-(1+4s^{2})^{-} \frac{1}{2}x^{a}$ . Thenequivalent to, where the last $repre\mathit{8}entati_{on}$ is defined by,$S’ \mathit{8}$$g_{s}$$g$$(R_{s}, \mathrm{L}_{g}^{2}(H^{a}))i\mathit{8}$(2.5)$R_{0}(X^{*})$$(R_{0}, \mathrm{L}_{gs}^{2}(H^{a}))$:$f(x^{a})\in \mathrm{L}_{g_{S}}^{2}(H^{a})-\sqrt{\frac{d(g_{s})_{x^{*}}}{dg_{s}}}(x^{a})f(X^{a}-x^{*})\in \mathrm{L}_{g_{s}}^{2}(H^{a})$(3) There exists another family of representations
property thatare inequivalent toare mutually inequivalent.
$(R_{\zeta_{c}}, \mathrm{L}_{g}^{2}(H^{a}))$$(R_{\zeta_{c}}, \mathrm{L}_{g}2(H^{a}))$$(R_{s}, \mathrm{L}_{g}^{2}(H^{a}))$for.

$(c\in \mathrm{R})$$dlc,$ $s\in$ R.

with theMoreover$(R_{\zeta_{\mathrm{c}}}, \mathrm{L}_{g}2(H^{a}))$The definition of$|x\in H\}(\subset \mathrm{L}_{g}^{2}(H^{a}))$is as follows. For any $h\in H^{*}$ we take a uniquesuch that$\zeta_{c}$$<x,$ $h>= \int_{H^{a}}<x,$ $x^{a}>Wh(Xa)g(dX^{a})$$W_{h}\in \mathrm{C}1\{<x,$ $x^{a}>$.

Put$\zeta_{\mathrm{C}}(X^{a}, h).:=\exp\{.\sqrt{-1}c\sum_{n=1}(W^{3}h_{n}(x-\varphi)-3Wh(_{X}n-\varphi)-W_{h_{n}}\mathrm{s}(X)+3Whn(X))\}\infty.\cdot$

For the further and detailed informations see [13].

119

3. 1-COCYCLES FOR ROTATIONALLY INVARIANT MEASURESIn this section we set up the following situation.
Let $H$ be a real separable Hilbert space ( $\dim(H)<\infty$ or $=\infty$ ),be a cylindrical$O(H)$ be a rotation group (-algebra on, if $\dim(H)=n<\infty$ ), andbe an $O(H)$ -quasi-invariant probability measure. Now consider a continuous 1-cocycle
defined on $H^{a}\mathrm{X}O(H)$ which satisfies the following conditions.
(1) For any fixed $U\in O(H),$is a -valued -measurable function.
.
(2) For any$U_{2}\in O(H)$ ,$\mathfrak{B}$$H^{a},$$\sigma$$O(H)=\mathrm{S}\mathrm{O}(\mathrm{n})$$\mu$$\theta$$S^{1}$$\theta(x^{a}, U)$$\mathfrak{B}$$\mathrm{J}$$U_{1},$$\theta(x^{a}, U_{1})\theta(tU_{1}x, U_{2}a)=\theta(Xa, U1U_{2})$for$\mu-\mathrm{a}.\mathrm{e}.x^{a}$.

(3) $\theta(x^{a}, U)arrow 1$ in , ifin the strong operator topology.
Such 1-cocycles arises in the representations of the semi-direct product of $H$ and $O(H)$ .
That is, let (V, ) andbe unitary representations of $H$ and $O(H)$ , respectively$\mu$$Uarrow \mathrm{I}\mathrm{d}$$(T,\mathcal{H})$$\mathcal{H}$which satisfy,(1)
is cyclic,(2)
is continuous on any finite dimensional subspace of $H$ andthe strong operator topology, and(3) for all $h\in H$ and for all $U\in O(H)$ ,
$V$
$V$

$T$

is continuous with

$T(U)V(h)=V(Uh)T(U)$ .

Then there exist an $O(H)$ -quasi-invariant probability measure on
and a continuous 1-cocycle for such that (V, ) andare realized as follows.
$\mu$$\theta$$\mathcal{H}$$\mu$$(H^{a}, \mathfrak{B})$$(T, \mathcal{H})$(3.1)$V(h)$:$f(x^{a})\in \mathrm{L}_{\mu}^{2}(H^{a})-\exp(\sqrt{-1}<h, x^{a}>)f(x^{a})\in \mathrm{L}_{\mu}^{2}(H^{a})$(3.2)$T(U)$:$f(x^{a})\in \mathrm{L}_{\mu}^{2}(H^{a})arrow\theta(x^{a}, U)\sqrt{\frac{d\mu_{U}}{d\mu}}(X^{a})f(^{ta}UX)\in \mathrm{L}_{\mu}^{2}(H^{a})$.
.

Moreover similar results with Theorem 2.2 also holds. We have only to change theergodic part to $O(H)$ -ergodic”. Thus the pair of representations (V, ) is also controledby the same two factors. However the situation is quite different from the previoue one.
First for the measure the following results are already known.
“$T$

Theorem 3.1. (1) For any rotationally quasi-invariant probability measure , there exists a rotationally invariant probability measure such that.
(2)
is represented as a supe $rp_{oS}ition$ of probability measures, whereisthe uniform measure on the sphere of radious centered at the origin, if $\dim(H)<\infty$ ,and is the centered $GaILSSian$ measure with variance , if $\dim(H)=\infty$ .
$\mu$

$\nu$

$\mu\simeq\nu$$\nu$$\{g_{C}\}_{c\in}[0,\infty)$$g_{c}$$c$$c^{2}$$g_{c}$For the proof, see [12] and [17]. Second the structure of 1-cocycles is very simple as isshown in the following theorem.

Theorem 3.2. Assume that $\dim(H)\neq 3$ . Then any continuovs 1-cocyde-coboundary. That is, there exists -valued -measurable function on
for each fixed $U\in O(H)$ ,$S^{1}$$\mathit{1}$$\mathfrak{B}$$\theta(x^{a}, U)=\frac{\phi(^{t}Ux^{a})}{\phi(x^{a})}$for p-a.

$e.x^{a}$.

$\phi$$\theta$$H^{a}$

for$\mu i\mathit{8}a$such that120

For the proof see [14]. From these theorems, we see that the pair of representation(V, ) is equivalent to the following one,
$T$

(3.3)

$V_{\nu}(h)$:$f(x^{a})\in \mathrm{L}_{\nu}^{2}(H^{a})arrow\exp(\sqrt{-1}<h,x^{a}>)f(x^{a})\in \mathrm{L}_{\nu}^{2}(H^{a})$(3.4)$T_{\nu}(U)$:$f(x^{a})\in \mathrm{L}_{\nu}^{2}(H^{a})arrow f(^{t}UX^{a})\in \mathrm{L}_{\nu}^{2}(H^{a})$,,and the equivalence (irreducibility) of the pair (V, ) defined by (3.3) and(3.4) are reduced to the equivalence of the corresponding rotationally invariant (ergodic) probability
measure , respectively. Further singIe representation of $O(H)$ ,
$T$

$\nu$

$R_{\theta}(U)$:$f(Xa)\in \mathrm{L}_{\mu}^{2}(H^{a})\mapsto\theta(x^{a}, U)\sqrt{\frac{d\mu_{U}}{d\mu}}(x^{a})f(tUX)a\in \mathrm{L}_{\mu}^{2}(H^{a})$is equivalent to the representation defined by (.3.4), and the properties for thesition are derived from the decomposition$\mathrm{d}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{e}\succ$$\dot{\mathrm{o}}\mathrm{f}$$\nu=\int_{[0,\infty)}g_{c}m(dC)$,and from the result for the irreducible
where $m$ is a Borel probability measure on, using multiple Wiener integrals $\mathcal{H}_{n}(n=0,1,2, \cdots)$ for
decomposition,
the Gaussian measure $g=g_{1}.$ Namely,$[0, \infty)$$\mathrm{L}_{g}^{2}(H^{a})=\Sigma\oplus \mathcal{H}_{n}$is completely reducible,
Theorem 3.3. Assume that $\dim(H)=\infty$ . Thenand as its irreducible components,(1) $(R_{g},\mathcal{H}_{n})(n=1,2, \cdots)$ appears
times in it and, according to $m(\mathrm{O})>0$ or(2)-times orappears$m(0)=0$ .
$(T_{\nu}, \mathrm{L}_{\mathcal{V}}^{2}(Ha))$$\dim(\mathrm{L}_{m}^{2})-$d.im$(R_{g}, \mathcal{H}0)$$(\mathrm{L}_{m}^{2})+1$$\dim(\mathrm{L}_{m}^{2})- times$

N.B. Here we give a counter example for Theorem 3.2, when $\dim(H)=3$ .
Let $e:=e_{3}={}^{t}(0,0,1)$ ,: $U\in SO(3)\mapsto Ue\in S^{2}$ and $N$ be a Borel cross section of$x\in S^{2}$. Then for any
and for any $U\in SO(3)$ there exists
such that$\mathcal{M}$$\tau\in \mathrm{R}$$\mathcal{M}$$U^{-1}N(x)=N(U^{-1}x)$ .
Put.
Then is a continuous 1-cocycle for the uniform measure on. However it is not a
1-coboundary. For the detailed informations this section, see [14].
$\theta(x, U):=\exp(\sqrt{-1}\tau)$$S^{2}$$\theta$$\mathrm{i}.\mathrm{n}$4. 1-COCYCLES ON THE GROUP OF DIFFEOMORPHISMS
Let $M=M^{d}$ be a paracompact ’-manifold andbe the set of all diffeomorphisms with compact supports. This section is a study of 1-cocycle on.
So let be a -finite smooth measure on $M$ which is locally equivalent to the Lebesgue, and take a canonical representation
measure onofsuch that$C^{\infty}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$\theta$

$g$

$\mu$

$M\cross \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$\sigma$$\mathrm{R}^{d}$$U_{\theta}$(4.1)$U_{\theta}(g)$:$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$f(P)\in \mathrm{L}_{\mu}^{2}(M)\mapsto\theta(P,g)\sqrt{\frac{d\mu_{g}}{d\mu}(P)}f(g^{-}(1P))\in \mathrm{L}_{\mu}^{2}(M)$,where is a continuous 1-cocycle. Here the topology onis the inductive limit$K\uparrow X$ , whereontopology ofis equipped with the topology
of uniform convergence with every derivative on the$\theta$$\tau$$\tau_{K}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(K):=\{g\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)|\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}g\subset K\}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(K),$$\tau_{K}$121

compact set $K$ . Exactly speaking, the continuity of is as follows. $\theta(P,g_{n})arrow 0$ ininand
if there exists some compact set $K$ such that. is never a group topology unless $M$ is compact. (See [16] in this issue). These. In this report we will
1-cocycles often appears in the representation theory ofgive some characterization of 1-cocycles which have much stronger continuousthan the original one. We anew give the defin,ition- of our present 1-cocycle .
$\theta$$\mu$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}g_{n}\subseteq K(n=1, \cdots)$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(K)$$g_{n}arrow \mathrm{I}\mathrm{d}$$\tau$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$\mathrm{p}.\mathrm{r},\mathrm{o}\mathrm{p}-.\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{y}$$\theta$

is said to be continuoIAs 1Definition 4.1. A-valued function on
cocycle, if and only if the following conditions are satisfied.
(1) For any,$S^{1}$$\theta$$g_{1},g_{2}\in.\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{o}(M.\cdot.

$M\cross \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$)$$\theta(P,g_{1})\theta(g^{-}11(P),g_{2})=\theta(P,g1g2)$(2) For each fixed $P\in M,$$\theta(P,g)$

to .

.

is a continuous function of$g\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}.(M)$with respect$\tau$

The analysis of continuous 1-cocycles is based on the following theorems.

Theorem 4.1. (Campbell–Hausdorff formula)
Let $X,$ $Y\in\Gamma_{0}(M)$ andfeomorphisms generated by $X,Y$ , respectively. Then$\{\mathrm{E}\mathrm{x}\mathrm{p}(tx)\}t\in \mathrm{R},$ $\{\mathrm{E}\mathrm{x}\mathrm{p}(tY)\}_{t}\in \mathrm{R}$

(1)
(2)

$\{\mathrm{E}\mathrm{x}\mathrm{p}(\frac{tX}{n})\circ \mathrm{E}\mathrm{x}\mathrm{p}(\frac{tY}{n})\}n$converges to$a\mathit{8}n$be 1-parameter subgroups of dif,tends$\mathrm{E}\mathrm{x}\mathrm{p}(t(X+Y))$$\{\mathrm{E}\mathrm{x}\mathrm{p}(-\frac{tX}{\sqrt{n}})\mathrm{o}\mathrm{E}\mathrm{x}\mathrm{p}(-\frac{tY}{\sqrt{n}})\circ \mathrm{E}\mathrm{X}\mathrm{p}(\frac{tX}{\sqrt{n}})\circ \mathrm{E}\mathrm{X}\mathrm{p}(\frac{tY}{\sqrt{n}})\}^{n}$

unifomly on every compact interval ofin.
containig
and$\tau_{K}$$\mathrm{t}o+\infty$$t_{J}$, andconverges to$\mathrm{E}\mathrm{x}\mathrm{p}(-t^{2}[x,Y])$

respectively, where $K$ is any compact set$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}Y$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}X$, where $X$ runs through all
Theorem 4.2. The group generated by-vectorwithacompact
supports,
dense
subsetwhich
theconnected
fieldsforms
of.
component of Id in$\tilde{G}$$C^{\infty}$$\mathrm{E}\mathrm{x}\mathrm{p}(x)$$i\mathit{8}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}^{*}(..M)$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$Using these theorems we restrict to the subgroup and analize it locally. Then, butmany lemmas are needed, the following results are obtained which is expected by T.Hirai
in the case of.
$\tilde{G}$$\theta$$M=\mathrm{R}^{d}$

Theorem 4.3. Assume that $M$ is simply connected. Then any continuous 1-cocycle
has the following canonical form,$\theta$$\theta(P,g)=\frac{c(g^{-1}(P))}{c(P)}(^{\frac{d\mu_{g}}{d\mu}(P})\mathrm{I}^{\sqrt{-1}}S\eta(g)$, whereis a -valued $continuo!L/S$ function on $M,$ is a real number anda $unitan/$, so it is a function
character on. (Actually, is a trivial character onon the discrete group
and are uniquely $dete7mined$ for , while
determined up to constant factors.
$S^{1}$$c$$s$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$\eta$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)/\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}^{*}(M).)S$$\eta i\mathit{8}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}^{*}(M)$$\theta$$\eta$$ci_{\mathit{8}}$Corollary 4.4. If $M$ is a compact Lie group, then the same holds for any continuous
1-cocycle .
$\theta$

If $M$ is not simply connected, then it is possible to exists a new 1-cocycle. For example, we have a following result. Let
in the caseand take$p=(u,z)\in$a continuous pathconnecting Id and . Then for each fixed, the second component of
has a continuous angular function.
$M=\mathrm{R}\cross S^{1}$$g\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(\mathrm{R}\cross S^{1})$$\{g_{t}\}_{0\leq}t\leq 1$$\mathrm{R}\cross S^{1}$$g$$g_{t}^{-1}(u, z)$$\theta(t, u, z)$122

The value $\varphi(u, z):=\theta(1,u, z)-\theta(\mathrm{O},u, z)$ only depends on $(g,u, z)$ and does not depend..
. So for any $\Omega\in[0,1)$ put
on a particular choice of.
$\zeta_{\Omega}((u, Z),g):=\exp(\sqrt{-1}\Omega\Psi(u, Z))$ .
...
$\{g_{t}\}_{0\leq}t\leq 1$’

and it is exten, ded to the whole groupThen
is a continuous 1-cocycle onin an essential unique way. We denote it again by .
$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(\mathrm{R}\cross S^{1})$$\zeta_{\Omega}$$\zeta_{\Omega}$

Theorem 4.5. If$M=\mathrm{R}\cross S^{1}$, the general form of$continuo’\llcorner \mathrm{A}s\mathit{1}$-cocycles$\theta(P,g)=\frac{c(g^{-1}(P))}{c(P)}(\frac{d\mu_{g}}{d\mu}(P)\mathrm{I}^{\sqrt{-1}}S\zeta_{\Omega}(P,g)\eta(g)$$i\mathit{8}$as follows,.

Any $\zeta_{\Omega}(0<\Omega<1)$ is never 1-cohomologus with any 1-cocycles appeared in Theorem4.3.
and are uniquely determined from and is dete $7mined$ up to constant factors.

$s,$$\Omega$$\theta$$c$$\eta$$\mathrm{L}\mathrm{a}s\mathrm{t}\mathrm{l}\mathrm{y}$, we shall list the following results with canonical representations defined by (4.1).

Theorem 4.6.
that $M$ is connected. Then.
(1) The representation
is irreducible for all continuous 1-cocycle .
:(2), if and onlyis equivalent toand are l-cohomologus.
$A_{\mathit{8}Su}me$$\mathrm{t}$$\theta$$(U_{\theta}, \mathrm{L}_{\mu}^{2}(M))$$(U_{\theta_{1}}, \mathrm{L}_{\mu}2(M))$$(U_{\theta_{2}}, \mathrm{L}_{\mu}2(M))$$if\theta_{1}$$\theta_{2}$For detailed informations in this section, see [15].
Department of Mathematics
Fukui University

REFERENCES[1] Gel’fand,I.M.,and Vilenkin,N.Ya., Generalized functions, vol.IV, Academic Press, (1961).
[2] Hirai,T., Construction of irreducible unitary representations of the infinite symmstric group,
J.Math.Kyoto Univ., 31 (1991), 495541.
[3] Hirai,T., Irreducible unitary representations of the group of diffeomorphisms of a non-compact
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0.fHe,rmit..ian

i.n:.finite$\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}..\cdot.\mathrm{m}.\mathrm{o}\mathrm{t}\mathrm{i}_{0}.\mathrm{n}.\mathrm{g}\mathrm{r}.\mathrm{o}\mathrm{u}\mathrm{p}$H..ans. Am..er..

522-536.

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[14] Shimomura,H., 1-cocycles for rotationally invariant measures, to appear.
[15] Shimomura,H., 1-cocycles on the group of diffeomorphisms, to appear.
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On group topologies
k\={o}ky\={u}roku, this issue.
$\mathrm{t}\mathrm{h}.$$|‘ \mathrm{o}\mathrm{n}$$\mathrm{s}_{\overline{\mathrm{u}}}\mathrm{r}\mathrm{i}\mathrm{k}\mathrm{a}\mathrm{i}\mathrm{S}\mathrm{e}\mathrm{k}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{y}\overline{\mathrm{u}}\mathrm{s}\mathrm{h}\mathrm{o}$e-$\mathrm{g}.\mathrm{r}.\mathrm{o}\mathrm{u}\mathrm{p}$$.

.

$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\infty_{\mathrm{i}}\mathrm{m}.$$\vee 0.\cdot \mathrm{f}$or.phisms,

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Usp.Mat.Nauk, 30 (1975), 3-50 ([19] Yamasaki,Y., Measures on infinite dimensional spaces, World Scientific, (1985).
$\mathrm{G}\mathrm{e}1’ \mathrm{f}\mathrm{a}\mathrm{n}\mathrm{d}_{:}\mathrm{I}.\mathrm{M}.$$=\mathrm{R}\mathrm{u}\mathrm{s}\mathrm{s}$