WIENER FUNCTIONALS ON SPACES OF LIE ÁLGEBRA VALUED 1-CURRENTS , AND UNITARY REPRESENTATIONS OF CURRENT GROUPS

JEAN MARION Well-known results about Brownian and Wiener functionals on abstract Wiener spaces are extended to Wiener functionals on the space of 9valued 1-currents on a manifold X, where 9 is the Lie algebra of a compact semisimple Lie group G. We introduce a family of Shigekawa-Sobolev spaces of generalized Wiener functionals and on each of them one gets a regular representation of the current group D(X,G) of G-valued and compactly supported smooth mappings on X. Then, a kind of Weyl's construction is used to associate, to each Riemannian flag of X, a ring of non located and order 1 unitary representations of D(X, G) including the energy representations studied in [23], [14] .


Introduction
If we should know a sufficiently large number of non trivial unitary representations of the current group D(X, G) it would be possible, as for finite dimensional Lie groups, to get a relevant non commutative harmonic analysis on it .Unfortunately, at the present time, our knowledge of the unitary dual of D(X, G) is rather poor, in spite of some efforts ( [12], [6], [23], [14]).
In these works a class of non located and order 1 continuous unitary representations of D(X, G) are constructed, often called energy representations, irreducible when dim (X) >_ 3; they are all realizable in a L'-space of Wiener functionals on the space of 9-valued 1-currents on the manifold X, where 9 is the Lie algebra of the compact semisimple Lie group G.
The original motivation of the present work, consisting of to find a consistent enlargement of the class of unitary representations of D(X, G) actually known, leads then to begin by a detailed study of these Wiener functionals .The section I is devoted to the construction of D(X, G)-invariant Euclidean structures on the space of 9-valued smooth 1-forms with compact support on X and Gaussian measures on the space Di (X, G) of G-valued 1-currents on X, associated with R.iemannian fiags.
The section II is devoted mainly to the study of the spaces of L2 -Wiener functionals on Di(X, G) in a way similar to the one used in [5] for the study of Brownian functionals associated to white noise, and the Wiener-Itó decomposition of such spaces is given.In section III we introduce a class of transforms, called M-transforms and which includes the Fourier-Wiener transform and the generalized Ornstein-Uhlenbeck semigroup ; the study of its infinitesimal generator leads to the realization of a Hilbert-Sobolev space of Wiener functionals .
In section IV we extend this result by the introduction for each real number a > 0, of a semigroup of contractions, (the Ornstein-Uhlenbeck semigroup corresponding to the case a = 1) whose infinitesimal generator allows to construct a Shigekawa's model of Hilbert-Sobolev space of Wiener functionals, genera-2,1 the senation described in [19], and denoted L .The section V is devoted to the construction and the study, for each a >_ 0, of a regular (of course unitary) representation of D(X, G) on LFá .A rather surprising fact, unknown in the case of finite dimensional Lie groups, is exhibited : on the subspace of polynomial Wiener functionals these various regular representation are equal and unitary with respect as well to the L2 scalar product as to the various scalar products LF a of Sobolev type.
The section VI is devoted to the construction of non located unitary representations of order 1 of D(X, G); it appears that they are unitarily equivalent to the energy representations constructed in [14] .
Lastly, in section VII, starting from the lot of representations constructed in section VI, and using a kind of generalization of the Weyl's construction, to each Riemannian flag of X is associated a ring of non located unitary representations of order 1 of D(X, G) called generalized energy representations, including as very particular elements the energy representations given in [14].

I. 9-valued 1-currents and energy Gaussian measures
In all this paper X is a Riemannian manifold with finite dimension do , and G is the Lie algebra of a compact semisimple Lie group G, endowed with the Euclidean structure coming from the opposite of its Killing form.Ad will denote the adjoint representation of G on G, N the set of non negative integers, and N* the set of strictly positive integers .
1.The nuclear space, of 9-valued 1-forms and its energy scalar products .a) Let DI (X, 9) be the space of all the 9-valued and compactly supported smooth 1-forms on X; with its Schwartz topology it is a countably Hilbert nuclear space .The nuclear Lie group D(X, G) of all the G-valued and compactly supported smooth mappings on X is a current group acting continuously on DI (X, 9) by a representation V such that for all g in D(X, G) and all w in DI (X, G), V(g)w in the 1-form : (1) x ~-4 (V(g)w)(x) = Adg (x) -w(x), x E X.
b) In order to endow Dl (X, 9) with a positive definite inner product for which the operators V(g) are unitary operators we shall need a geometrical object studied and used in [14], called a Riemannian flag (see also [13] for a summary of results) .
Definition 1.A Riemannian flag of X is a collection F = (Xk,rk)o<k<P, with 0 _< p :5 do, such that: -for each integer k with 0 < k < p, Xk is a connected submanifold of X with dimension dok, and rk is a R.iemannian metric of class Cl on Xk ; -XP C . ..CXkCXk-1 C . ..CXo =X .Let F = (Xk,rk)o<k<P be a Riemannian flag of X ; for each integer k with 0 < k < p let us denote by dxk the volume measure on Xk, and ( , )k,,, the scalar product on the tangent space T--Xk of Xk at x with respect to the Riemannian metric rk.One gets a scalar product (, )k on D1(Xk, g), defined for all .1,.1' in DI (Xk,9) by: (A, A% = f tr (1 \"(x) --X(x»dxk, Xk where tr is the trace operator, and where `denotes here the adjoint operation according to the Euclidean structures on T=Xk and 9. c) The energy scalar product < , > F on D1 (X, 9) associated to the Riemannian flag F is then given by taking, for all w, w' in D1(X, g): where wk and w' are the restrictions of w and w' respectively to the submanifold Xk, 0 < k < p.We shall denote by HF the Hilbert completion of D1 (X, G) with respect to < , >F, and by 1 1 F the corresponding norm.
Owing to the unitarity of the adjoint representation Ad of G on 9, it follows that for all g in D(X, G) the operators V(g) extend into unitary operators on H F , and then: Lemma 1.For any Riemannian fiag F of X, V is a continuous unitary representation of ¡he current group D(X, G) on H F .
Of course, V is highly reducible, looking like a continuous sum of representations of G on 9.
2. The space of g-valued 1-currents and its energy Gaussian measures .a) Let Di (X, 9) be the dual space of D1 (X, G); it is the space of g-valued 1-currents on X.In the present work it ,is the basic space on which the distribution of a generaliaed stochastic process will be located in order to expanda relevant analysis .
A well-known and standard result about dual spaces of countably Hilbert nuclear spaces allows to claim .thatfor any Riemannian flag F of X, the triple [Dl(X, G), HF, D* (X, g)] is a Gelf'and triple, Le. an Hilbertian triad in the sense of [3].We shall denote by < , > the pairing of Dl (X, G) and its dual space.
The natural extension V* of the representation V on Di (X, 9) is such that for all g in D(X, G), all X in Di (X, G), and all w in Dl (X,,9) : (4) < V *(g)X,w >=< X,V(g-1)w > .
b) Notice that for each Riemannian flag F of X the mapping CF Dl (X, 9) -> R given by : Moreover, p,F satisfies the condition: is a characteristic functionaL From the Bochner-Minlos theorem it follows that there exists one and only one Gaussian measure MF on Di (X, JC) with mean zero, and Fourier transform PF given by : J < x, W > -< X, w' > dMF(X) =< w, w' >F for all w, w' in Dl (X, G) .MF will be called the energy Gaussian measure associated with the Riemannian flag F. c) Notice that two Riemannian flags: F = (Xk, rk)o<k<p and F' = (Xk, rk)o<k<p , are equal if and only if p = p', and for all k = 0,1, . . .,p, the Riemannian manifolds (Xk, rk) and (Xk, rk) are equal.Now let us recall the result of Feldman and Hajek assertiog that two Gaussian measures are either equivalent or disjoint .
As a direct consequence of proposition II-2 given in [14] one gets: Proposition 1 .The energy Gaussian measures UF and PF', associated with two different Riemannian flags F and F' are disjoint .
d) Now a crucial point is the behaviour of MF through the action of the current group D(X, G) by the representation V*.Proposition 2. Let B be the topological a-field of Di (X, 9), and let F be a Riemannian flag of X.For any element g in D(X,G), V*(g) is a Bmeasurable mapping from D*(X, 9) onto itself.Moreover MF o V*(g) = pF .
Proof.The first part of the assertion follows from the fact that V is a continuous unitary representation of D(X, G) on HF .In order to prove the second part it suffices to compute the characteristic functional of the measure M F o V*(g) and to see that it is equal to the characteristic functional CF of hF : by the unitary of the operators V(g).Therefore MF is D(X, G)-invariant .
II.The space of L'-Wiener functionals on the 9-valued 1-currents 1.The spaces LP(Di(X, g); h ; MF) .Let F be a Riemannian flag of X, let B F be the completion of the topological a-field B of Di (X, 9) (with respect to MF), let h be a separable Hilbert space with norm II IIh and topological a-field Bh .
Definition 2. The measurable mappings from the measurable space Di (X, G), BF) into the measurable space (h, Bh) are called h-valued Wiener functionals on Di (X, C) .
The above definition generalizes in a natural way the concept of Brownian functional studied in [5], [10], [17] .
Of course, two h-valued Wiener functionals are considered in the same class whenever they are equal p F-almost everywhere ; so there is no difficulty to define for all real p > 1 the space LP(Di (X, C) ; h; {UF) of h-valued Wiener functionals ob such that the mapping X F--+ ]I'¿(X)IIh 1s MF-integrable, the corresponding norm being given by (g) , (X,4) In this section we are mainly interested by the Hilbert spaces LZ(D*(X,G) ;h ; MF) and more particularly by L2 (Dl (X, g) ; C ; MF) which will be more simply denoted LF ; its elements will be called L 2 -Wiener functionals .
2. Polynomial Wiener functionals.Formula (7) shows that for any 1-form w in D1 (X, 9) the mapping X H< X, w > is a Gaussian random variable with mean zero and variante Iw1F ; therefore, for any positive integer n one has (9) I < X'w > ¡In : PF(X) = (2n)!(n!)-1 -2 -n ' IWI F ; D1 (X,4) it follows that the mapping X i-->< X, w >n belongs to LF for all positive integer n, and all element w in Di (X, G).Now, using the Schwarz inequality and a standard argument, one easily proves that for any collection (W1, . ..,wq) of elements in Di(X, 9) and any collection (ni, . . ., nq ) of non negative integers the mapping X b-> < X, W1 >n, . . .< X,wq >nq belongs to LF .
We notice, from the above discussion, that the space P of all the C-valued polynomial Wiener functionals is an algebra contained in LF .
Remark.Let h be a separable Hilbert space; the elements of P ® h are the h-valued polynomial Wiener functionals on Di(X, C) .
A Riemannian flag F of X being given, let us consider a complete orthonormal system (Wi)iEN-of HF with all the Wi in DI (X, G) .Owing to the above discussion and from the density of Di (X, 9) in (H F)* = HF it follows easily that: Proposition 3.For all positive integer p, P ® h is a dense linear subspace of LP(Di(X, g) ; h; MF) .In particular ¡he algebra P is dense in LF .

Exponential Wiener functionals.
a) Definition 4: A C-valued Wiener functional 4> is said an exponential Wiener functional if there exist a complex number a and a 1-form w in DI (X, ~) such that P(X) = exp [a < X, w >] for all X in Di (X, G) .Lemma 2. All ¡he exponential Wiener functionals are L2 -Wiener functionals.
Proof.The assertion follows from the fact that: exp [a < X, w >] _ En_o ñ, < X,w >n, and that the mappings X ~< X, w >n belong to LF .
b) To each w in D1 (X, 9) let us associate the exponential Wiener functional w : X 1--> exp [i < X,w >] .We notice that PF(w) = PF(W) = CF(w), and that the complex vector space E spanned by all the w is an algebra .More precisely : Proposition 4 .The algebra E is dense in LF .
Proof.-It is similar to the one given for exponential Brownian functionals given in [5], theorem 4.1 .By lemma 2 we have E C L2 F .Now let us recall that 13 denotes the topological Q-field of Di (X, 9) and that we have selected a complete orthonormal system (Wk)kEN-in H F, the Wk being in DI(X,G).
For each positive integer n, let us consider the sub-Q-field of B generated by the functionals X H< X,wk >, 1 _< k <_ n, and denoted Bn .(Bn)nEN " is an increasing sequence converging to B.
In order to prove the assertion it suffices to show that any element ~¿ in LF which is orthogonal to all the elements of the form X H IIk=1 exp [ itk < X, wk >j for all positive integer n, and all t1, . . ., tn in R, is necessarily zero MF -almost everywhere.
So, let us assume that for any positive integer n and any sequence (t1, . . ., tn) of real numbers the Wiener functional -P satisfies : and let us denote by E(1IBn) the conditional expectation of 1 with respect to the sub-u-field Bn.One has : The Fourier transform of X v-r E(P(X)IBn) is then zero on the subspace spanned by (w1, . ..,wn) ; therefore, for all positive integer n, E(¿D¡Bn) is zero MF-a.e. (Bn)n being increasing sequence converging to the Q-field B, it follows that E(PIB) = OP = 0, MF-a .e .The assertion is then proved .
4 .The Wiener-Itó decomposition of L'F .a) Let us recall first a well-known result: the system of mappings ( nl -Hn )nEN, where Hn denotes the Hermite polynomial function given by: Let A be the set of sequences v = (vk)kEN " of elements of N such that vk = 0 except for finitely many k's.For such an element v = (vk)kEN " in A we will denote : (12 ) ¡Vi = En1vk ; v! = r1k1 Vk!For all n in N, An will denote the set {v E n/IvI = n} .Now a Riemannian flag F of X being given, we select a complete orthonormal system (Wk)kEN-of HF , with all the wk in D1 (X, G) .
Definition 5: For all v in A, the mapping h defined on Di (X, G) by: (13 ) h,,(X) = HH,k (< X,wk >) is called a Fourier-Hermite polynomial Wiener functional .
We have to give now the Wiener chaos decomposition of LF ; such a decomposition is well-known in the case of the space of L2-Brownian functionals ([51) and in the case of L2-functionals on an abstract Wiener space ([191).
Proposition 5.The collection ( v!h ) E,\ is a complete orthonormal system in LF .For each n in N le¡ us denote by Zñ the closed subspace of LF spanned by the family ( v!h,),,En.; one has the orthogonal direct sum, called the Wiener-Itó decomposition : The proof is standard and we omit it .Of cocarse ¡he Wiener-Itó decomposition of LF is independent of a particular choice of the orthogonal system (Wk)kEN-of HF .
Remark: Let us consider the symmetric Fock space S(HF ) constructed with HF as one particle space: where the sum is taken with orthogonal components, and where (H F)On,s denotes the space of symmetric n-tensors on HF .As it is well-known there is a natural Hilbert isomorphism from S(HF) onto LF ([4]).It is easy to see that for each integer n, this isomorphism restricts to a Hilbert isomorphism from For a complex number z :~0 we shall denote by -,ízthe complex number such that ( .,íz -)  = z and such that arg ( ,~íz -) is in [0, 7r] .For each non zero complex number z we define the transform MZ from L1 (Di(X, g) ; C ; M F) into itself by : (14) Mzq(X) = J $(zX + 1z 2 0)dMF (e),
From the definition of Fourier-Hermite polynomials and from the above formula it follows that: Lemma 3 .Let F be a Riemannian flag of X, and let z be any non zero complex number.For all v in An, n E N one has: MF h v = znhv .
Notice that MF is a linear operator on LF .As a direct consequence of lemma 3 one gets : Lemma 4 .Let F and z be as in lemma 3.One has: In particular, for each integer n we have ¡he commutation: M F F F .MF.
z ' pn = Pn z 2 .The Fourier-Wiener transform.Definition 6.Let F be a Riemannian flag of X ; the M-transform MF is called the Fourier-Wiener transform associated to F, and denoted JF.
Let us compute now the Fourier-Wiener transform of the basic Wiener functionals ; we recall that for w in D l (X, G), w denotes the exponential Wiener functional X v-+ exp [i < X, w >] .
Proposition 7. Let n be in N, let w be in Di (X,9) and leí v be in An .
Proof.-(i) is a trivial consequence of proposition 5 and lemma 3.
(ii) For all X in Di (X, G) one has: An immediate consequence of lemma 4, proposition 7 (i), and the definitions Corollary.(i) For all n in N, p* ,7 F = ~F P F n ; (ii) for any non zero complex number z one has: 3 .F-Ornstein-Uhlenbeck semigroup and operator .Let F be a Riemannian flag of X, and let us consider the family (Fi)t>o of M-transforma defined for all real numbers t > 0 by : (17) Ft = Mé t .
Notice that from lemma 3, for all integers n > 0 and all v in nn one has: (19) Fth = e-nth, t > 0.
Notice that usual Ornstein-Uhlenbeck semigroups in infinite dimension are defined on Lp-spaces based on abstract Wiener spaces (see e.g.[19]).
The computation of the infinitesimal generator LF of (Ft)t>o, called the F-Ornstein-Uhlenbeck operator is very easy : It follows that : Remark.The space P of polynomial Wiener functionals is contained in D(L F) because for a Fourier-Hermite polynomial Wiener functional h , v E A, Pnh,, is zero if ¡vi :~n and is equal to h if ¡vi = n ; as jiv! hjL2 = 1 it follOWS that : E' 1 n2 11Pn h.¡¡i2 = Ivi2(v!)-1 < +oo .
Therefore, using standard arguments, from the above remark and proposition 9 it follows that : Proposition 10.LF eatends to a Hermitian operator with domain contai- ning the algebraic sum Eñ=o Zñ, and for each integer n >_ 0, Zñ the eigenspace corresponding to the eigenvalue -n .
Proposition 11.LF eatends to the eaponential Wiener functionals; moreover, for all z in C and all w in D1(X, JC) one has : for all X in Di (X, CG) .
1.The Hilbert-Sobolev space LFl .Let F be a Riemannian flag of X; the Ornstein-Uhlenbeck operator L F = -E°°o npñ is a negative definite self-adjoint operator on LF with domain containing the dense subspace P of LF consisting of the polynomial Wiener functionals.
Let us consider the operator CF , ' defined on P by : with domain D(LF,«) : The operator CF,1 looks like the Cauchy operator defined as the square root of the opposite of the usual Laplace-Beltrami operator.It allows to define a new scalar product < , >L2,1 on P by : (24) < 'M >L',i=< <I>, Y' >L2 + < CF'1 .,p,C F,1 0 >L2 .
Of course, <, >L2 and <, >L2,1 are not equivalent because C F,1 is unbounded, although they are equivalent on each subspace p (1 n .Formally, the scalar product given by (24) looks like the classical Sobolev scalar product of L 2,1 type.As it is well-known, in the case of Wiener functionals on an abstract Wiener space of infinite dimension there are two ways in order to define frames of Sobolev type.The one uses Fréchet derivatives and leads to the Shigekawa's model of Sobolev space ; the other, using stochastic Gateaux derivatives, leads to the Kusuoka-Stroock's model of Sobolev space ( [18], [11]).In fact it is proved in [20], theorem 3 .1,that these two ways lead to the same Sobolev spaces .In the generalization that we shall give in the next section of such spaces we cannot claim that the two ways lead to the same spaces .
We shall denote by L2,1 the Hilbert completion of P with respect to the scalar product < , >L2,1 .
2 .The Shigekawa-Hilbert-Sobolev spaces LF. ' We generalize here the situation described in 1) .We select a Riemannian flag F of X.Each element a in the set R+* of strictly positive real numbers gives rise to a contraction semigroup (Ft )t>o on L2F, with F¿ given by: F°= E°°e -n°t F t n=0 pn' Notice that the F-Ornstein-Uhlenbeck semigroup given in definition 7 corresponds to the case a = 1.The corresponding infinitesimal generator FF,' is given by: Notice that the operators LF, ' are unbounded, negative definite, and selfadjoint on L2 .Moreover: Lemma 5.For any a in R+*, D(LF,") contains the space P of polynomial Wiener functionals and also ¡he exponential Wiener functionals .
Proof. .Let v be in A, and let n be in N; then pn (h) = 0 if Ivi ~n, and Pn'(h,) = h if ¡vi = n.It follows that E°°_1 n2ajipn (h )jjL Z = w1 2°(vl) -1 by proposition 5. Therefore h belongs to D(LF, °) and then p C D(LF, ") .Now the same argument that the one used in proposition 11 allows to claim that LF,a extends to exponential Wiener functionals.
Let us consider now the square root CF,a of the positive definite selfadjoint operator -LF,a : (28) This generalized Cauchy operator allows to get a new scalar product <, >L«2,i on P by : (29) with ~¿, 0 in P.
Proof.The assertion follows from the fact that, for a given v in A, if we suppose that a' = a + 0 with ,Q > 0, then, with n = Jv1, < CF,a+0 v! h , CF°a +0 v! h >L 2= n a+Q = no < CF,a v! h , CF'a v! h >L2 .
Definition 8.The Hilbert completion of P with respect to <, >L2 , 1, denoted L2, a, will be called a Shigekawa-Hilbert-Sobolev space of order 1 on Di(X, G).
Notice that if we should have taken a = 0, the corresponding Hilbert space would be L2, o = LF .
V. The regular representations RF of the current group D(X, G) 1 .The regular representation Ró on LF .Let F be a Riemannian flag of X .In I-2, proposition 2 we have seen that the action of the current group D(X, G) on Di (X, 9) given by the representation V* leaves invariant the energy Gaussian measure MF .It follows that for all g in D(X, G) the operators Ró (g) defined for all <P in LF by : (Ró(g)<p)(X) = <p(V*(g-1)X), X E D*(X, G), are unitary operators on LF = LFá ; RF : g F-> Rá(g) is then a continuous unitary representation of D(X, G) on LF,O .
Definition 9.The unitary representation Ró will be called the regular representation of D(X, G) on LF.Proposition 12.For all n in N the closed subspace Zñ is invariant by Ró and for all g in D(X, G) : Ró (g) .PF = pñ Ró (g) Proof.-Let w be in Dl (X, G), and let q, be the polynomial Wiener functional X ~-4< X, w >; for all g in D(X,G), Ró(g)q,(X) =< V *(g -1 )X,w > =< X, V(g)w >= gv(y),(X).It follows that each Fourier-Hermite polynomial Wiener functional is transformed by Ró (g) into a Fourier-Hermite polynomial Wiener functional with same degree .Therefore, from proposition 5 it follows that Zñ is invariant by Ró (g); this proves the assertion.
As a direct consequence of proposition 12 one gets : Corollary 1.Let cp be any mapping from N into R and let Tw be the operator: Tw = En=pep(n)pñ'.For all g in D(X, G) one has the commutation: In particular: On the other side, from corollary 2 of proposition 12 one has: < CF,« -RF(g),p, CF'a -RF (g)o >Lz = < RF (g) -CF'04, RF (g) .
CF,a0 >Lz ** < CF,c, .p, CF'a , 0 >L2 From the definition of <, > L !,i, and from (*) and (**) it follows therefore that : Remarks .On P fl ZÍ , Le .on the space of polynomial Wiener functionals of degree 1, CF'a restricts to the identity so that the completion of P U Zi with respect to the scalar product < , > L2,1 is exactly Zi, for all a in R+* ; therefore Rá and Ra F. are equal on Z1.This is, of course, no more true for subspaces P f1 Zñ, with n > 1 .
2) To my knowledge, there is no equivalent phenomenon for regular representations of finite dimensional Lie groups .
(32) < RF(g), P, RF (g)V >L2,1 < (D,10 > L2,1 . Consequently one easily concludes that RF extends to a continuous unitary representation of D(X, G) on L2,1 .VI. Elementary energy representations of the current group D(X, G) 1. On unitary action of Dl (X,9) on LF .a) Let F be a Riemannian flag of X, and for all w in Dl (X, 9) let us consider the translation Tw on Di (X, G) : X ~4 X +w, which is such that for all 9 in Dl (X, G), < Tj X, 6 >=< X, B > -F < w, 8 >F .A well-known result about quasi-invariance of Gaussian measures by translations (see e.g.[9], [5]), allows to assert that MF is quasi-invariant by Tj, the corresponding Radon-Nikodym derivative being given by : dMF(X) An immediate consequence of this is the following: Proposition 14 .For each w in Di (X, 9) ¡he mapping UW defined for all We have then, for all g in D(X, G): a(g) = « F F (g) j,1 >)4 = exp (-2 1 dg -g-1 jF) = s(g), which prove the assertion .We notice that the equivalence of 7r F with (FF)®a implies that when the energy representation 7r F is irreductible, then rF is necessarily irreducible .
Definition 10 .The unitary representations rF, where F is any Riemannian flag of X, will be called the elementary energy representations of D(X, G) .
VII .The rings of generalized energy representations of D(X, G) 1 .Motivations .At the present time the only non located unitary representations of D(X, G) we know and which are not trivially reducible are the elementary energy representations .The present section is devoted to the enlargement of such representations in order to get a ring of unitary, non located an order 1 (in the sense of [23], [12]) representations of the current group D(X, G) including the elementary energy representations as basic elements .A motivation for this enlargement comes from two points of view.On one hand it is necessary to have sufficiently non trivial and non located unitary representations of D(X, G) in hope to get a consistent non commutative harmonic analysis on spaces of generalized Wiener functionals ; on the other hand we can also hope that such representation constitute a relevant tool in order to study G-valued stochastic variables and stochastic G-valued multiplicative measures ([1]).
Our construction is similar, in some sense, to the Weyl's -construction of irreducible finite dimensional representations of the general linear group and its generalization given in [7], [22] for finite functional dimension representations of the group of diffeomorphism of a non compact manifold.
2 .The generalized energy representation Iñ p .Let F be a Riemannian flag of X with associated energy Gaussian measure hF on Di (X, G) .For each positive integer n we shall denote by pF the product measure pF x x p,F (n copies) on the product space Di (X, g)n, and by Sn the symmetric group of permutations of a set having n elements ; Sn will denote the set of (classes of) unitary representations of S n.We have then a fibered set S = U Sn consisting of pairs (n, p) where n is a positive integer and p a class n<1 of unitary equivalence of unitary representations of Sn in some Hilbert space EP .