CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS CORRESPONDING TO PRODUCT MEASURES OF GROUP OF FINITE UPPER-TRIANGULAR MATRICES. A. V.

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1 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS CORRESPONDING TO PRODUCT MEASURES OF GROUP OF FINITE UPPER-TRIANGULAR MATRICES A. V. Kosyak Received: Abstract. We define the analog of the regular representations of the group of finite upper-triangular matrices of infinite order corresponding to quasi-invariant product measures on the group of all upper-triangular matrices and give the criterium of irreducibility of constructed representations under some technical conditions. In, 2 the criterium was proved for an arbitrary centered Gaussian product-measures on the same group and in 3 for groups of the interval and circle diffeomorphisms.. Regular representation. Apparently, an analog of the regular representation of infinite-dimensional groups (current group) appears firstly in 4, 5, 6. In 7 reducibility of the analog of the regular representation of current group, corresponding to the Wiener measure and the commutation theorem was proved. An analog of the regular representation for any infinite-dimensional group G, using G quasi-invariant measures on some complitions G of such a group is defined in, 2. Let B0 be the group of finite upper-triangular matrices of infinite order, B be the group of all such matrices (not necessary finite), b be its Lie algebra B 0 {I + x I + k<n x kn E kn x is finite}, B {I + x I + k<n x kn E kn x is arbitrary}, b {x k<n x kn E kn x is arbitrary} where E kn, k, n <, are matrix units of infinite order. On the algebra b we define a measure µ as infinite tensor product of probabilty measures µ kn, k, n N on R µ(x) k<n µ kn (x kn ) 99 Mathematics Subject Classification. 22E65. Key words and phrases. Infinite dimensional group, regular representations, Gaussian measures. Typeset by AMS-TEX

2 2 A. V. KOSYAK and let dx, x R be the Lebesgue measure on R. Define the measure µ ρ on the group B as the image of the measure µ with respect to the bijective mapping ρ : b x ρ(x) I + x B. In the sequal we will use the same notation µ for both measures on B and b. Let ν be any probability measure on the group B and ν n be its projection to the subgroup B(n, R) {I + x B0 I + x I + k<m n x kme km } of the group B0 (in the case of tensor product we have µ n (x) k<m n µ km (x km )). Let also dχ n (x) k<m n dx km be the Haar measure on the group B(n, R). Let us denote by R and L the right and the left action of the group B on itself: R s (t) ts, L s (t) st, t, s B and let µ Rt, µ Lt, t B0 are imagies of the measure µ with respect to the right and the left action of the group B0 on B. For an arbitrary probability measure ν we have Lemma. ν Rt ν, t B 0 ν n χ n n N. For an arbitrary product measure µ k<n µ kn we have Lemma 2. µ Rt µ, t B 0 dµ kn (x kn ) dx kn k < n, i.e. dµ kn (x kn ) µ kn (x kn )dx kn, µ kn (x kn ) 0 almost everywhere (mod dx kn ). Let us denote by M kn (p) x p µ kn (x)dx, R Mkn (p) ((i D kn (µ)) p, ) L 2 (R, µ kn ), p, 2,..., where D kn (µ) / x kn + / x kn (lnµ /2 kn (x kn)). In the case when the Fourier transform Fµ /2 kn (x) of the function µ/2 kn (y) is positive, we define In this case we have also (see (9)) Fµ /2 kn (x) (2π) /2 exp(ixy)µ /2 kn (y)dy, R µ kn (x) Fµ /2 kn (x) 2. () ((i D kn (µ)) p, ) M kn (p) x p µ kn (x)dx. R Lemma 3. µ Lt µ, t B 0 S L kn(µ) mn+ M km (2)M nm (2) <, k < n.

3 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS... 3 Lemma 4. For k < n we have µ LI+tE kn µ, t R \0 Skn L (µ). If µ Rt µ and µ Lt µ, t B0 one can define (see 2) an analog of the right and the left regular representation of the group B0 lim B(n, R), T R,µ, T L,µ : B 0 U(H µ L 2 (B, dµ)), corresponding to a B 0 quasi-invariant measure µ on the group B (2) (3) (T R,µ t f)(x) (dµ(xt)/dµ(x)) /2 f(xt), (Ts L,µ f)(x) (dµ(s x)/dµ(x)) /2 f(s x). For the generators A R,µ kn (AL,µ kn ) of the one-parameter groups I+tE kn, t R, k < n, corresponding to the right T R,µ (respectively left T L,µ ) regular representation we have the following formulas: (4) (5) A R,µ kn d k dt T R,µ t0 I+tE kn x rk D rn (µ) + D kn (µ), r A L,µ kn d dt T L,µ I+tE kn t0 (D kn (µ) + mn+ x nm D km (µ)). 2. Irreducibility. Let the family of measures (µ kn ) k<n have the following properties: ) sup n,n>k Mkn (2)M kn (2) c k <, k 2, 3,... 2) sup n,n>k Mkn (4)( M kn (2)) 2 d k <, n, 2,... Theorem. Let conditions )-2) hold for the measure µ k<n µ kn. In this case the right regular representation T R,µ of the group B0 is irreducible if and only if no left actions are admissible for the measure µ µ Lt µ, t B \e. Remark. The statement of the theorem (without conditions )-2)) was expresed by R. S. Ismagilov as the conjecture for Gaussian product measures dµ b (x) k<n (b kn /π) /2 exp( b kn x 2 kn)dx kn on the group B 0 of finite upper-triangular matrices of infinite order and was proved in, 2. We note that in the case of Gaussian measures conditions )-2) hold automatically. Proof of the Lemma and 2.. Since for any n 2, 3,... the group B(n, R) acts transitivily on itself the condition ν Rt ν, t B0 is equivalent to the following one νn Rt ν n, t B(n, R) for large n. But any G quasi-invariant measure on a locally compact group G is equivalent with the Haar measure on this group, so the last condition is equivalent to ν n χ n, n 2, 3,... We recall the definition and properties of the Hellinger integral (8, Chap. 2, 2). Suppose that µ and ν are two probability measures on the measure space

4 4 A. V. KOSYAK (X, B). Assume that λ is probability measure such that µ λ, ν λ for example λ (µ + ν)/2. The Hellinger integral H(µ, ν) of two measures µ and ν is defined as follows: dµ dν H(µ, ν) dλ dλ dλ. It does not depend on λ and has the following properties: X (H) 0 H(µ, ν) (the Schwartz inequality); (H2) H(µ, ν) µ ν; (H3) H(µ, ν) 0 µ ν; (H4) µ ν H(µ, ν) > 0. The converse to (H4) does not hold in general. Proof of the Lemma 3. Since the one-dimensional subgroups I+tE kn, t R, k < n generate the group B0 we have µ Lt µ, t B 0 µ LI+tE kn µ, k < n, t R. Since µ and µ LI+tE kn are product measures and x kn x kn+... x km... (I + te kn )(I + x) (I + te kn ) x nn+... x nm x kn + t x kn+ + tx nn+... x km + tx nm x nn+... x nm the condition µ LI+tE kn µ, k < n is equivalent by crirerium of Kakutani 9 (see also 0, 6, Theorem ) with H(µ LI+tE kn, µ) > 0, t 0. We have H(µ LI+tE kn, µ) H(µ L I+tEkn kn, µ kn ) R mn+ H((µ km µ nm ) LI+tE kn, (µkm µ nm )) ( ) /2 dµkn (x kn + t) µ kn(x kn)dx kn dµ kn (x kn ) ( ) /2 dµkm (x km + tx nm ) µ km(x km)µ nm(x nm)dx km dx nm dµ nm (x nm ) mn+ R 2 (exp(td kn (µ)), ) mn+ Let µ LI+tE kn µ k < n, t R. (exp(tx nm D km (µ)), ).

5 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS... 5 We see that so, using, Lemma 3, we have The direct calculation gives us H(µ LI+tE kn, µ) (exp( ta L,µ kn ), ) H µ H(µ LI+tE kn, µ) > 0, t 0 A L,µ kn H µ <. A L,µ kn 2 H µ M kn (2) + mn+ M km (2)M nm (2). Proof of the Lemma 4. Since the measures µ and µ LI+tE kn we have proved also are product measures µ LI+tE kn µ, t R \0, k < n S L kn(µ), k < n. Proof of the Theorem. Necessity is obvious since in the oposite case we have for some id : e t 0 B nontrivial operator T L,µ t 0 that commute with the right regular representation. Sufficiency. Let µ Lt µ, t B \e hence µ LI+tE kn µ, t R \0, k < n, then by Lemma 4 Skn L (µ), k < n. We will show in this case that by generators A R,µ kn we will be able to approximate the operators of multiplication by independent variables (x kn ) k<n. We can use the direct calculation. In the case when the Fourier transform Fµ /2 kn (x), k < n is positive to approximate variables (x kn) k<n we can use also some Fourier-Wiener transform F m (see 2) for diagonalizing the commutative family of the operators (A R,µ kn ) k m<n (see formula (0) below). For any m 2, 3,... we define the partial Fourier-Wiener transform F µ m between two spaces H µ L 2 (B, dµ) H m H Am H m, and H m,µ H m H Am H m, where H m L 2 (B m, dµ m ), B m B(m, R), µ m k<n m µ kn, H Am L 2 (A m, dµ Am ), A m {I + x kn E kn }, µ Am k m<n µ kn, k m<n H m L 2 (B m, dµ m ), B m {I + x kn E kn }, µ m m<k<n µ kn, m<k<n H Am L 2 (A m, d µ Am ), µ Am k m<n µ kn.

6 6 A. V. KOSYAK Let us denote by F µ kn the measure dµ kn (x kn ) the one-dimensional Fourier transform, corresponding to µ /2 kn (x kn)u µ kn L 2 (R, dx) F L 2 (R, dµ kn ) F µ kn By definition F µ kn (U µ kn ) FU µ kn, where so we have (6) (F µ kn f)(y kn) µ /2 kn L 2 (R, dy). U µ kn µ/2 kn (y kn) L 2 (R, d µ kn ) (Ff)(y) 2π R exp(iyx)f(x)dx, (y kn) f(x kn )exp(iy kn x kn )µ /2 2π kn (x kn)dx kn. R Obviously, F µ kn, where L2 (R, µ kn ) is the function (x), x R. Let us define (7) F m µ k m<n F µ kn, m 2, 3,... Lemma 5 (see 2). F µ m, m 2, 3,... is an isometrie between two spaces H µ L 2 (B dµ) H m H Am H m, and H m,µ H m H Am H m. Proof of the equality (). Since the Fourier-image of the operator i d/dy is the operator of the multiplication by the independent variable y i.e. Fi d/dy(f) y, and U µ kn D kn(µ)(u µ kn ) / y, we have (8) F µ kn i D kn (µ)(f µ kn ) y kn. Indeed F µ kn i D kn (µ)(f µ kn ) (U µ kn ) FU µ kn i D kn (µ)(u µ kn ) F U µ kn (U µ kn ) Fi / y kn F U µ kn y kn. Let us denote by H(µ kn ) L 2 (R, dµ kn ), H( µ kn ) L 2 (R, d µ kn ). Operator F µ kn (U µ kn ) FU µ kn is a unitary operator: F µ kn U(H(µ kn), H( µ kn )) since U µ kn, F and U µ kn are unitary operators. So we have, since F µ kn ((i D kn (µ)) p, ) H(µkn ) ((i D kn (µ)) p (F µ kn ), (F µ kn ) ) H(µkn ) (F µ kn (i D kn (µ)) p (F µ kn ), ) H( µkn ) (y p kn, ) H( µ kn ) y kn µ p kn(y kn )dy kn, R (9) ((i D kn (µ)) p, ) H(µkn ) y kn µ p kn(y kn )dy kn. R

7 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS... 7 Using (4),(7) and (8) for k m < n we have (0) Ã R,µ kn k (F m µ )AR,µ kn (F m µ ) x rk F rn µ D rn(µ)(f rn µ ) + F µ kn D kn(µ)(f µ kn ), Ã R,µ kn r k i( x rk y rn + y kn ). r Definition. Recall 3 that a non necessary bounded self-adjoint operator A in the Hilbert space H is affiliated to the von-neumann algebra M of operators in this Hilbert space H (notation Aη M) if exp(ita) M t R. Let us denote by A R,µ the von-neumann algebra, generated by the right regular representation: A R,µ (T R,µ t t B0 ). Let us denote also by < f n n, 2,... > the closure of the linear space, generated by the set of vectors {f n } n in a Hilbert space H. We prove Lemma 6 8 assuming that the conditions ) and 2) hold. Lemma 6. x 2 η A R,µ if S2 L (µ). Moreover x 2 < A R,µ n AR,µ 2n 3 n >, if and only if S2 L (µ). In this case we have also D n(µ), D 2n+ (µ)η A R,L,b, n 2, 3,... Proof. Let us estimate δ 2,3,m min ( m t n A R,µ n {t AR,µ 2n + x 2) 2 H µ. n} We can do it directly. Since (see (4)) A R,µ n D n(µ), A R,µ 2n x 2D n (µ) + D 2n (µ), 2 < n, so A R,µ n AR,µ 2n x 2D n (µ) 2 +D n (µ)d 2n (µ), hence we have if m t n M n (2) m t n A R,µ n AR,µ 2n + x 2 2 H µ m t n (x 2 Dn(µ) 2 + D n (µ)d 2n (µ)) + x 2 2 H µ m t n x 2 (Dn 2 (µ) + M n (2)) + D n (µ)d 2n (µ) 2 H µ m m m x 2 2 (D 2 n(µ) + M n (2)) 2 + D n (µ) 2 D 2n (µ) 2 M 2 (2)( M n (4) M n 2 (2)) + M n (2) M 2n (2) ( M n (4) M n 2 (2)) + M n (2) M n (2).

8 8 A. V. KOSYAK Using Fourier-Wiener transform, if it is possible, we will have the same answer. Indeed, for m 2 we have (see (0)) So ÃR,µ n ÃR,µ 2n à R,µ n iy n, à R,µ 2n i(x 2y n + y 2n ), 2 < n. (x 2yn 2 + y n y 2n ), hence we have if m t n M n (2), m t n à R,b n ÃR,b 2n + x 2 2 H 2,µ m t n (x 2 yn 2 + y n y 2n ) x 2 2 H 2,µ m t n (x 2 (yn 2 M n (2)) + y n y 2n ) 2 H 2,µ m m m x 2 2 y 2 n M n (2) 2 + y n 2 y 2n 2 M 2 (2)( M n (4) M n(2)) 2 + M n (2) M 2n (2) ( M n (4) M n 2 (2)) + M n (2) M n (2). To estimate the last expression we use the following equality (see 2, p. 25) () min { m a n {t n} n n m t n b n } m n, b 2 n a n see also 4, 26 bibliography, p. 79 min { m x 2 nc n {t n} n m x n } m n. c n n Using () we have if m t n M n (2), m δ 2,3,m min t n A R,µ t n AR,µ 2n + x 2 2 H µ n m M 2 n (2) ( M n(4) M 2 n (2))+ M n(2) M 2n(2) Σ 2,3,m. So lim m δ 2,3,m 0 if and only if Σ 2,3,. Using ) and 2) we have Σ 2,3, ( M n(4) M 2 n (2) ) + M 2n(2) M n(2) M n (2) M 2n (2) : S 2 L (µ) c 2 M n (2)M 2n (2) S2 L (µ).

9 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS... 9 Finally we have x 2 η A R,µ and D n (µ), D 2n+ (µ) A R,µ 2n+ x 2D n+ (µ) η A R,µ, n 2. We prove that we have convergence A m,m 2 m 2 in the strong resolvent sence. This means that m t n i A R,µ n i A R,µ 2n x 2, if m 2 exp(ita m,m 2 ) exp(itx 2 ), t R. By Theorem VIII.25 of 5, it suffices to show the convergence A m,m 2 f x 2 f for any f D, where D is a common essential domain for all the operators A m,m 2 and A x 2. For the role of D we choose a dense set consisting of finite linear combinations of arbitrary monomials x α x α2 2 xα3 xα23... xαn xα2n..., α ij 0,,..., i < j. Obviously D is the common essential domain for the operators A m,m 2 and A, since D consists of analytic vectors for the operators A m,m 2 and A. The function f D is cylindrical, so for some m this function f does not depend on the variables x n,..., x n n for n m and we have f f m. Since the operator A m,m 2 does not act on the variables x n,..., x n n for n < m, sauf on x 2, we have Indeed, m 2 m2 min ( {t n} min ( t n A R,µ nm m 2 nm t n nm t n A R,µ {t n} ( m2 n AR,µ n AR,µ nm t n A R,µ 2n + x 2)f 2 H µ n AR,µ 2n + x 2)f m 2 H µ 2n + x 2) 2 H µ 0 if m 2. x 2 (Dn 2 (µ) + M n (2)) + D n (µ)d 2n (µ) f m 2 H µ m 2 m 2 x 2 f m t n (Dn(µ) 2 + M n (2)) + f m D n (µ)d 2n (µ)) 2 H µ nm nm m x 2 f m 2 t n (Dn 2 (µ) + M n (2)) 2 m 2 + f m 2 t n D n (µ)d 2n (µ) 2 H µ nm m x 2 2 t n (Dn(µ) 2 + M n (2)) 2 + m 2 ( t n A R,µ n AR,µ 2n + x 2) 2 H µ. nm m 2 nm t n D n (µ)d 2n (µ) 2 H µ

10 0 A. V. KOSYAK In addition t n A n m,m 2 f t n A n m <,m 2 < for some t > 0. n! n! n0 n0 This means that it suffices to prove that is analytic vector for the operators A m,m 2. It is evident. Analogously, any vector f D is analytic for the operator A. So we have proved if S L 2 (µ) x 2 η A R,µ and D n (µ), D 2n+ (µ)η A R,µ, 2 n. Lemma 7. x 3, x 23 η A R,µ if S3 L (µ) and SL 23 (µ). In this case we have also D 3n (µ), 3 < n. Proof. Let us estimate Since we have δ 3,4,m min ( m t n A R,µ n {t AR,µ 3n + x 3) 2 H µ. n} A R,µ n D n(µ), A R,µ 3n x 3D n (µ) + x 23 D 2n (µ) + D 3n (µ), 3 < n A R,µ n AR,µ 3n x 3D 2 n (µ) + x 23D n (µ)d 2n (µ) + D n (µ)d 3n (µ), D 2n (µ)a R,µ 3n x 3D n (µ)d 2n (µ) + x 23 D 2 2n (µ) + D 2n(µ)D 3n (µ), hence we have if m t n M n (2), m t n A R,µ n AR,µ 3n + x 3 2 H µ m t n (x 3 Dn(µ) 2 + x 23 D n (µ)d 2n (µ) + D n (µ)d 3n (µ)) + x 3 2 H µ m t n (x 3 (Dn(µ) 2 + M n (2)) + x 23 D n (µ)d 2n (µ) + D n (µ)d 3n (µ)) 2 H µ m x 3 2 (D 2 n (µ) + M n (2)) 2 + x 23 2 D n (µ) 2 D 2n (µ) 2 + D n (µ) 2 D 3n (µ) 2 m m M 3 (2)( M n (4) M n 2 (2)) + M 23(2) M n (2) M 2n (2) + M n (2) M 3n (2) ( M n (4) M n 2 (2)) + M n (2) M 2n (2) + M n (2) M 3n (2).

11 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS... Using () we have if m t n M n (2), m δ 3,4,m min t n A R,µ t n AR,µ 3n + x 3 2 H µ n m M 2 n (2) ( M n(4) M 2 n (2))+ M n(2) M 2n(2)+ M n(2) M 3n(2) : (Σ 3,4,m ). So lim m δ 3,4,m 0 if and only if Σ 3,4,, and using 2) we have Σ 3,4, ( M n(4) M ) + M 2n(2) n 2 (2) M + M 3n(2) n(2) M n(2) M n (2) (2) M 2n (2) + M 3n (2) : σ 3(µ). If m t n M 2n (2), we have m t n D 2n (µ)a R,µ 3n + x 23 2 H µ m t n (x 3 D n (µ)d 2n (µ) + x 23 D2n 2 (µ) + D 2n(µ)D 3n (µ)) + x 23 2 H µ m t n (x 3 D n (µ)d 2n (µ) + x 23 (D2n 2 (µ) + M 2n (2)) + D 2n (µ)d 3n (µ)) 2 H µ m x 3 2 D n (µ) 2 D 2n (µ) 2 + x 23 2 (D 2 2n(µ) + M 2n (2)) 2 + D 2n (µ) 2 D 3n (µ) 2 m m M 3 (2) M n (2) M 2n (2) + M 23 (2)( M 2n (4) M 2n(2)) 2 + M 2n (2) M 3n (2) Mn (2) M 2n (2) + ( M 2n (4) M 2n(2)) 2 + M 2n (2) M 3n (2). Using () we have if m t n M 2n (2), m δ 23,4,m min t n A R,µ t n AR,µ 3n + x 23 2 H µ n m M 2 2n (2) M n(2) M 2n(2)+( M 2n(4) M 2 2n (2))+ M 2n(2) M 3n(2) : (Σ 23,4,m ). So lim m δ 23,4,m 0 if and only if Σ 23,4, and we have, using 2) Σ 23,4, M n(2) M + ( M 2n(4) 2n(2) M ) + M 3n(2) 2n 2 (2) M 2n(2) M 2n (2) (3) M n (2) + M 3n (2) : σ 23(µ).

12 2 A. V. KOSYAK Using ) we have S 3(µ) L : S L 23 (µ) : M n (2) M 3n (2) (c 3) M 2n (2) M 3n (2) (c 3) M n (2)M 3n (2) (c 3 ) S L 3(µ), M 2n (2)M 3n (2) (c 3 ) S23 L (µ). Hence by Lemma 9 one of the series σ 3 (µ) or σ 23 (µ) is divergent. Let σ 3 (µ) so x 3 η A R,µ and x 23 < D 2n (µ)(a R,µ 3n ( M 2n(4) M 2 2n (2) ) + M 3n(2) M 2n(2) x 3D n (µ)) 4 < n > M 2n (2) M 3n (2) S 23(µ) L, so x 23 η A R,µ. If σ 23 (µ) we have x 23 η A R,µ and x 3 < D n (µ)(a R,µ 3n ( M n(4) M 2 n (2) ) + M 3n(2) M n(2) x 23D 2n (µ)) 4 < n > M n (2) M 3n (2) S 3 L (µ). so x 3 η A R,µ. Finally we have x 3, x 23 η A R,µ and D 3n (µ) η A R,µ, 3 < n, since D 3n (µ) A R,µ 3n x 3D n (µ) x 23 D 2n (µ), 3 < n. Lemma 8. {x rp+ } r<p+ η A R,µ if Srp+ L (µ), r < p+. In this case we have also {D p+s (µ)} p+<s η A R,µ. Proof. Let us estimate for some r, r p δ rp+,m,m 2 min {t n} ( m2 t n D rn (µ)a R,µ p+n + x rp+) 2 H µ. nm We have (see (4)): D rn (µ)a R,µ p+n D rn(µ)( p x sp+ D sn (µ) + D p+n (µ)), s

13 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS... 3 hence, if m 2 nm t n Mrn (2), we have m2 δ rp+,m,m 2 min min m2 {t n} min {t n} m2 {t n} t n D rn (µ)( nm D rn (µ)( nm t n t n D rn (µ)a R,µ p+n + x rp+ 2 H µ nm p x sp+ D sn (µ) + D p+n (µ)) + x rp+ 2 H µ s + x rp+ (D rn (µ) 2 + M rn (2)) min m 2 {t n} nm Mrn (2)( p s,s r p s,s r + M rp+ (2)( M rn (4) M 2 rn(2)) 2 H µ x sp+ D sn (µ) + D p+n (µ)) M sp+ (2) M sn (2) + M p+n (2)) m2 M2 rn (2) nm M p rn(2)( s,s r Msp+(2) M sn(2)+ M p+n(2))+m rp+(2)( M rn(4) M rn 2 (2)) (Σ rp+,m,m 2 ). Σ rp+,p+2, np+2 p np+2 s,s r M sn(2) M rn(2) + M p+n(2) M rn(2) M rn (2) p+ M s,s r sn (2) : σ rp+(µ). + ( M rn(4) ) M rn 2 (2) Lemma 9. Let ) 2) holds and Srp+ L (µ), r p. Then for some r, r p we have Proof. Using ) we have S rp+ L (µ) : σ rp+ (µ) np+ np+2 M rn (2) p+ M s,s r sn (2). M rn (2) M p+n (2) (c p+) np+2 (c p+ ) Srp+ L (µ), r p. so the proof follows from the Lemma 2.5, p. 256 in 2. M rn (2)M p+n (2) Let for some r we have σ rp+(µ). In this case we can aproximate the variable x rp+. To aproximate x rp+ for r r we use anouther combination D rn (µ)(a R,µ p+n x r p+d rn(µ)) D rn (µ)( p s,s r x sp+ D sn (µ) + D p+n (µ)).

14 4 A. V. KOSYAK We have if m 2 nm t n Mrn (2) δ r rp+,m,m 2 min min {t n} min {t n} m2 m 2 {t n} m2 t n D rn (µ)(a R,µ p+n x r p+d rn(µ)) + x rp+ 2 H µ nm p 2 H µ t n D rn (µ)( x sp+ D sn (µ) + D p+n (µ)) + x rp+ nm s,s r p D rn (µ)( x sp+ D sn (µ) + D p+n (µ)) nm t n + x rp+ (D 2 rn (µ) + M rn (2)) min m 2 {t n} nm Mrn (2)( s,s {r,r } 2 H µ p s,s {r,r } + M rp+ (2)( M rn (4) M 2 rn (2)) M sp+ (2) M sn (2) + M p+n (2)) m2 M2 rn (2) nm M p rn(2)( s,s {r,r } Msp+(2) M sn(2)+ M p+n(2))+m rp+(2)( M rn(4) M rn 2 (2)) (Σ r rp+,m,m 2 ). Σ r rp+,p+2, np+2 p np+2 s,s {r,r } p+ s,s {r,r } M rn (2) M sn (2) M sn(2) M rn(2) + M p+n(2) M rn(2) : σr rp+ (µ). By Lemma 9 for some r r, r p we have σ r rp+ (µ) np+2 + ( M rn(4) ) M rn 2 (2) M rn (2) p+ M s,s {r,r } sn (2). Let σ r r 2p+ (µ), then x r 2p+ η A R,µ and so on. At the end we have {x rp+ } r<p+ η A R,µ and {D p+n (µ)} p+<n η A R,µ since p D p+n (µ) A R,µ p+n x rp+ D rn (µ) η A R,µ, p + < n. r Finally we have {x kn } k<n η A R,µ.

15 B 0 CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS... 5 Let now a bounded operator A L(H µ ) commute with T R,µ t : A, T R,µ t 0, t then A, exp(itx kn ) 0, t R, k < n, so the operator A is an operator of multiplication in the space H µ by some function: A a(x). Since a(x), T R,µ t 0 t B 0, the function a(x) is B 0 -right invariant: a(xt) a(x) t B 0 so by ergodicity of the measure µ we have a(x) const. The Theorem is proved. Corollary. Let conditions )-2) hold for the measure µ. In this case three following conditions are equivalent: i) the representation T R,µ is irreducible, ii) µ Lt µ, t B \e, iii) Skn L (µ), k < n. Proof. By the Theorem we have: iii) i) so i) ii) iii) i). References. Kosyak, A. V., Irreducibility criterion for regular Gaussian representations of group of finite upper triangular matrices, Funct. Anal. i Priloz. 24 (990), no. 3, (Russian) 2. Kosyak, A. V., Criteria for irreducibility and equivalence of regular Gaussian representations of group of finite upper-triangular matrices of infinite order, Selecta. Math. Soviet. (992), Kosyak, A. V., Irreducible regular Gaussian representations of the group of the interval and circle diffeomorphisms, J. Funct. Anal. 25 (994), Albeverio, S. and Hoegh-Krohn, R., The energy representation of Sobolev-Lie group, Preprint, University Bielefeld, Ismagilov, R. S., Representations of the group of smooth mappings of a segment in a compact Lie group, Funct. Anal. i Priloz. 5 (98), no. 2, (Russian) 6. Albeverio, S., Hoegh-Krohn, R., and Testard, D., Irreducibility and reducibility for the energy representation of a group of mappings of a Riemannian manifold into a compact Lie group, J. Funct. Anal. 4 (98), Albeverio, S., Hoegh-Krohn, R., Testard, D., and Vershik, A., Factorial representations of path groups, J. Funct. Anal. 5 (983), Kuo, H. H., Gaussian measures in Banach spaces, Lecture Notes in Math., 463, Springer, Berlin, Kakutani, S., On equivalence of infinite product measures, Ann. Math. 4 (948), no. 9, Skorokhod, A. V., Integration in Hilbert space, Springer, Berlin, Kosyak, A. V., Measures on group of upper-trianguler matrices of infinite order quasi-invariant with respect to inverse mapping, Funct. Anal. i Priloz. 34 (2000), no., (Russian) 2. Kosyak, A. V. and Zekri, R., Regular representations of infinite-dimensional groups and factors, I., Methods Funct. Anal. Topology 6 (2000), no. 2, Dixmier, J., Les algèbres d operateurs dans l espace hilbertien, 2 e édition, Gautirs-Villars, Paris, Beckenbach, E. F. and Bellman, R., Inequalities, Springer-Verlag, Berlin, Göttingen, Heidelberg, Reed, M. and Simon, B., Methods of Modern Mathematical Physics. I, Academic Press, New York, 972. Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs ka, Kyiv, 060, Ukraine address: kosyak@imath.kiev.ua