Seminar Sophus Lie 3 (1993) 177–184 Version of July 15, 1995

Irreducibility of Gaussian regular representations of a group of germs of real analytic diffeomorphisms

Frank Leitenberger

1. Introduction

The regular representation of a finite dimensional Lie group decomposes into a direct sum of all unitary representations (compact case) or into a direct integral of almost all unitary representations with respect to the Plancherel measure (non-compact case, see [W]). Therefore it is natural to study the situation in the case of infinite dimensional groups. Due to A. Weil there is no Haar measure on groups which are not locally compact, but it is possible to construct an analogue of the regular representation for infinite dimensional Lie groups. The left regular representation is defined on H = L_{2}(\overline{G}, dμ) by

T (ϕ)f(ψ) := \sqrt{\frac{dμ(L_{ϕ^{−1}}ψ)}{dμ(ψ)}}f(L_{ϕ^{−1}}ψ), ϕ ∈ G, ψ ∈ \overline{G},

where \overline{G} is a topological group which contains G as a dense subset, dμ is a measure on \overline{G} quasinvariant under G (see [Os]) and L_{ϕ} is the left action.

Contrarily to the locally compact case in the case of infinite dimensional groups it is possible that the regular representation is irreducible for certain measures dμ. (This is impossible in the locally compact case because the operators of the right and the left regular representation commute.) First such results were obtained by N.I. Nessonov and A.V. Kosyak for the group of finite uppertriangular matrices of infinite order with units on the diagonal (see [K1], [K2]). This group is an inductive limit of finite dimensional Lie groups. It arises the question if such an result is possible for (weak) projective limits of finite dimensional Lie groups.

This article is devoted to the irreducibility of certain Gaussian regular representations of a diffeomorphism group of germs of the real line. The group is a subgroup of the group of lower-triangular matrices with units on the diagonal and a dense subgroup of the projective limit of the finite dimensional groups of n-Jets.

The proof is based on the fact that the operators x_{i}, ∂_{i} + m_{i}x_{i}, i ∈ ℕ of an irreducible representation of the infinite Heisenberg algebra in L_{2}(ℝ^{∞}, dμ_{m}) may be approximated by the generators of one-parameter subgroups of G.

This article is organized as follows: in section 2 we present some facts about the group G. In section 3 the analogue of a left regular representation is discussed. These representations depend on on a certain infinite dimensional measure dμ_{m} quasiinvariant under G. We give a characterization of all diagonal Gaussian measures quasiinvariant under G. In section 4 the generators of a class of oneparameter subgroups of the left regular representation are calculated. In section 5 we present the idea of the proof of the irreducibility for certain measures dμ_{m}.

2. Preliminaries

Our purpose in this section is to introduce the basic objects of the consideration.

Definition 2.1. By G we call the group of all real analytic transformations ϕ(x) of neighbourhoods of zero of the real line with ϕ(0) = 0 and ϕ′(0) = 1, that is,

G := {ϕ(x) = x + a_{1}x^{2} + a_{2}x^{3} + ⋯; a_{n} ∈ R; lim sup |a_{n}|^{1/n} < ∞}

equipped with the multiplication rule ϕ ∘ ψ(x) := ψ(ϕ(x)).

Endow G with a topological structure: consider the vector spaces G_{R}, (R > 0)

G_{R} = {ϕ(x) = a_{0}x + a_{1}x^{2} + ⋯; a_{n} ∈ R; lim sup |a_{n}|^{1/n} < 1/R}

and the seminorm system

|ϕ|_{r} := sup_{|x|<r} |ϕ(x)|; r = R(1 − 1/n); n > 1

or the equivalent seminorm system (see [K])

|ϕ|_{M} := sup_{k≥1} |a_{k}M^{k}|; M < R.

Then G_{R} is a Frechet space. Equip ⋃_{R>0} G_{R} with the inductive limit topology. Identify G with the hyperplane a_{0} = 1. Then the group G is a topological space.

Proposition 2.1. G is a topological group.

3. The definition of the left regular representation

Consider the embedding of G into the group

F = {ψ(x) = x + y_{1}x^{2} + y_{2}x^{3} + ⋯ | y_{i} ∈ R}

of formal power series: G ⊂ F ≃ ℝ^{∞} , and the left action of G on F

L_{ϕ} ψ(x) := ϕ ∘ ψ(x) = ψ(ϕ(x)), ϕ ∈ G, ψ ∈ F.

L_{ϕ} may be rewritten in matrix form

L_{ϕ}ψ = L ( a_{1} ; a_{2} ; a_{3} ; ⋮ ) ( y_{1} ; y_{2} ; y_{3} ; ⋮ ) = ( \begin{matrix}1 & 0 & 0 & ⋯ \\ a_{12} & 1 & 0 & ⋯ \\ a_{13} & a_{12} & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ... \end{matrix} ) ( y_{1} ; y_{2} ; y_{3} ; ⋮ ) + ( a_{1} ; a_{2} ; a_{3} ; ⋮ )

= Ay + b, where
a_{ij} = ∑_{{i_{1}+i_{2}+···+i_{j+1}=i+1 \atop i_{1},···,i_{j+1}>0}} a_{i_{1}−1} ⋯ a_{i_{j+1}−1}, a_{0} := 1.

This is obtained by a comparison of the coefficients of the power series ϕ, ψ and ϕ ∘ ψ. The L_{ϕ} are affine maps of F ≃ ℝ^{∞}.

For sequences of positive real numbers m = (m_{n})_{n≥1} on F Gaussian diagonal measures are defined:

dμ_{m} = ⨂_{n=1}^{∞} \sqrt{\frac{m_{n}}{π}} e^{−m_{n}y_{n}^{2}}dy_{n} .

In the following measures dμ_{m} are considered where the positive sequence m = (m_{n})_{n≥1} satisfies the condition (∗):

(∗)  ∑_{n=1}^{∞}(n + 1)^{2} \frac{m_{n+1}}{m_{n}} < ∞.

Lemma 3.1. Suppose that m = (m_{n})_{n≥1} satisfies the condition (∗). Then

(i) ∃C > 0 : m_{n} < C^{n}/(n!)^{2}.

(ii) ∀M > 0 : ∑_{k∈ℕ} m_{n}M^{n} < ∞.

(iii) ∀M ∈ ℕ : ∑_{i,n∈ℕ} \binom{n+i}{n}^{2}M^{2i} \frac{m_{n+i}}{m_{n}} < ∞.

Proof. (i): Condition (∗) implies that it exists a C′ > 0 with \frac{m_{n+1}}{m_{n}} (n + 1)^{2} < C′, that is, m_{n+1} < m_{n}C′/(n + 1)^{2} and from this by successively inserting we get m_{n+1} < m_{1}C^{′n}/(n + 1)!^{2}. The proposition follows with C := max{C′, m_{1}}. (ii) turns out from (i). (iii):

∑_{i=1}^{∞} ∑_{n=1}^{∞} \binom{n + i}{n}^{2}M^{2i} \frac{m_{n+i}}{m_{n}}

= ∑_{i=1}^{∞} \frac{1}{i!^{2}} ∑_{n=1}^{∞}(n + 1)^{2}(n + 2)^{2} ⋯ (n + i)^{2} \frac{m_{n+i}}{m_{n+i−1}} ⋯ \frac{m_{n+2}}{m_{n+1}} \frac{m_{n+1}}{m_{n}} M^{2i}

≤ ∑_{i=1}^{∞} \frac{1}{i!^{2}} ∑_{n_{1},n_{2},···,n_{i}=1}^{∞} (\frac{(n_{1} + 1)^{2}...(n_{i} + 1)^{2}m_{n_{1}+1}...m_{n_{i}+1}}{m_{n_{1}}m_{n_{2}} ⋯ m_{n_{i}}} ) M^{2i}

= ∑_{i=1}^{∞} \frac{1}{i!^{2}} (∑_{n=1}^{∞}(n + 1)^{2} \frac{m_{n+1}}{m_{n}})^{i}M^{2i} = f(M^{2} ∑_{n=1}^{∞}(n + 1)^{2} \frac{m_{n+1}}{m_{n}}) < ∞ ,

where f(x) = ∑_{i=1}^{∞} \frac{x^{i}}{i!^{2}} is a power series with an infinite radius of convergence.

Let the image of the measure dμ_{m} under the map L_{ϕ}, ϕ ∈ G be denoted by dμ_{m}^{L_{ϕ}}. The next proposition is devoted to a characterization of all diagonal Gaussian measures quasiinvariant under G.

Proposition 3.2. The measures dμ_{m} and dμ_{m}^{L_{ϕ}} are equivalent for all ϕ ∈ G if and only if they satisfy the condition (∗).

Proof. According to [S-F] the Gaussian measures dμ_{m} and dμ_{m}^{Ay+b} are equivalent if and only if Mb ∈ ℓ_{2} and MAM^{−1} − I is a Hilbert-Schmidt operator. Here is

M = ( \begin{matrix}\sqrt{m_{1}} & 0 & 0 & ⋯ \\ 0 & \sqrt{m_{2}} & 0 & ⋯ \\ 0 & 0 & \sqrt{m_{3}} & ⋯ \\ ⋮ & ⋮ & ⋮ & ... \end{matrix} ) .

1.((∗) is necessary): When (1) is taken into account in particular a_{n+1,n} = (n + 1)a_{1} is obtained. It follows for the Hilbert-Schmidt norm

|MAM^{−1} − I|_{2}^{2} = ∑_{j<i} a_{ij}^{2} \frac{m_{i}}{m_{j}} ≥ ∑_{n=1}^{∞} a_{n+1,n}^{2} \frac{m_{n+1}}{m_{n}}

= ∑_{n=1}^{∞}(n + 1)^{2}a_{1}^{2} \frac{m_{n+1}}{m_{n}} = a_{1}^{2} ∑_{n=1}^{∞}(n + 1)^{2} \frac{m_{n+1}}{m_{n}} .

Hence we get the proposition choosing ϕ ∈ G with a_{1} ≠ 0.

2.((∗) is sufficient): Because ϕ = x+a_{1}x^{2} +a_{2}x^{3} +⋯ ∈ G is a power series with a finite radius of convergence there exists a M > 0 with |a_{i}| ≤ M^{i}. We find from (1) that

|a_{ij}| ≤ ∑_{{i_{1}+···+i_{j+1}=i+1 \atop i_{1},···,i_{j+1}>0}}^{∞} M^{i−j} = \binom{i}{j}M^{i−j} .

Applying Lemma 3.1(iii) we obtain

|MAM^{−1} − I|_{2}^{2} = ∑_{j<i} a_{ij}^{2} \frac{m_{i}}{m_{j}} = ∑_{i=1}^{∞} ∑_{n=1}^{∞} a_{n+i,n}^{2} \frac{m_{n+i}}{m_{n}}

≤ ∑_{i=1}^{∞} ∑_{n=1}^{∞} \binom{n + i}{n}^{2}M^{2i} \frac{m_{n+i}}{m_{n}} < ∞ .

Moreover, we have Mb ∈ ℓ_{2}, since |a_{i}| < M^{i} and Lemma 3.1(ii).

Because of Proposition 3.2 the following definition makes sense:

Definition 3.1. Let m = (m_{n})_{n≥1} be a positive real sequence satisfying (∗). Then on H = L_{2}(F, dμ_{m}) the left regular representation is defined by

T_{m}(ϕ)f(ψ) := \sqrt{\frac{dμ_{m}(L_{ϕ^{−1}}ψ)}{dμ_{m}(ψ)}}f(L_{ϕ^{−1}}ψ).

Remark. Definition 3.1 allows the following generalization to topological groups B, which are not locally compact. Instead of B consider the triple (B′, B, dμ). Here B′ is a topological group, which contains B as a dense subgroup and dμ is a quasiinvariant measure under the left action of B on B′ (see [Os]). The left regular representation is defined by

T_{dμ}^{B′,B}(g_{0})f(g) := \sqrt{\frac{dμ_{m}(L_{g_{0}^{−1}}g)}{dμ_{m}(g)}}f(L_{g_{0}^{−1}}g), g ∈ B′, g_{0} ∈ B,

in particular T_{m} = T_{dμ_{m}}^{F,G}.

Proposition 3.3. The representation T_{m} is unitary and continuous.

4. The derived representation of the left regular representation

Our purpose in this subsection is to calculate the generators of some oneparameter subgroups of G in the left regular representation.

An easy calculation shows:

Lemma 4.1. The subsets G_{n} = \{ \frac{x}{\sqrt[n]{1−tnx^{n}}} : t ∈ R\} ⊂ G; n ∈ ℕ form oneparameter subgroups.

Let A_{n} the selfadjoint operator which generates the one-parameter subgroup T_{m}(G_{n}) of the left regular representation (see [R-S]). Further consider the in H dense set D = lin{⋃_{p≥1} C_{0}^{∞}(ℝ^{p})} of continuations of C^{∞}-functions with compact support in ℝ^{p} on ℝ^{∞}.

Proposition 4.2. There is D ⊂ D(A_{n}) and

A_{n}|_{D}f = − (∑_{i=0}^{∞}(i + 1)y_{i} (\frac{∂}{∂_{i+n}} − m_{i+n}y_{i+n})) f, y_{0} := 1.

Remark. The domain D is not invariant under the action of A_{n}.

5. Irreducibility

Consider the left regular representation T_{m}. According to Proposition 3.2 T_{m} is well defined for sequences m = (m_{n})_{n≥1} satisfying condition (∗), that is, ∑_{k=1}^{∞}(i + 1)^{2} \frac{m_{i+1}}{m_{i}} < ∞.

Theorem 5.1. T_{m} is irreducible for sufficiently fastly decreasing sequences (m_{n})_{n≥1}. Indication of Proof. Let W be the von Neumann algebra generated by the operators T_{m}(ϕ), ϕ ∈ G. We have to show that W′ = {λI}.

The self-adjoint operators A_{n} are affiliated to the subgroups generated by themselves and therefore they are also affiliated to W.

For n ≥ 1 consider the sequence of symmetric operators

C_{n,k} := \frac{−1}{(n + 1)m_{n+k}} (A_{n+k}A_{k} + A_{k}A_{n+k}), k ∈ ℕ.

Define C_{n} as the weak graph limit of the C_{n,k}, i.e., C_{n}ϕ := h, if there is a sequence ϕ_{k} → ϕ, ϕ_{k} ∈ D(C_{n,k}) such that ⟨C_{n,k}ϕ_{k}, g⟩ ⟶_{k→∞} ⟨h, g⟩ ∀g ∈ H.

Lemma 5.2. (i) lim_{k→∞}⟨C_{n,k}f, g⟩ = ⟨y_{n}f, g⟩ for f ∈ D, g ∈ H. (ii) \overline{C_{n}} is equal to the selfadjoint multiplication operator y_{n}.

Since the A_{n} are affiliated to W, the A_{n+k}A_{k} + A_{k}A_{n+k} commute with W on D, because

CA_{l}A_{k}f = A_{l}CA_{k}f = A_{l}A_{k}Cf

for f ∈ D, C ∈ W′. That is,

⟨(A_{n+k}A_{k} + A_{k}A_{n+k})f, C^{∗}g⟩ = ⟨(A_{k+n}A_{k} + A_{k}A_{n+k})Cf, g⟩,

where f ∈ D(C_{n}), g ∈ H. Multiplying both sides with \frac{−1}{(n+1)m_{n+k}} and taking the limit for k → ∞ one obtains according to Lemma 5.2(i):

⟨C_{n}f, C^{∗}g⟩ = ⟨CC_{n}f, g⟩ = ⟨C_{n}Cf, g⟩ ∀g ∈ H,

that is,

CC_{n}f = C_{n}Cf, f ∈ D.

We take the closure and conclude because of Lemma 5.2(ii) and the essentially selfadjointness of y_{n}|_{D}

Cy_{n} ⊆ y_{n}C,

i.e., y_{n} is affiliated to W.

Next we want to show that the operators

D_{n} := \overline{\frac{∂}{∂y_{n}} − m_{n}y_{n}}

are also affiliated to W.

Consider the sequence

A_{n}|_{D} = D_{n} + 2y_{1}D_{n+1} + 3y_{2}D_{n+2} + 4y_{3}D_{n+3} + ⋯
A_{n+1}|_{D} =  D_{n+1}  + 2y_{1}D_{n+2} + 3y_{2}D_{n+3} + ⋯
A_{n+2}|_{D} =  D_{n+2}  + 2y_{1}D_{n+3} + ⋯
A_{n+3}|_{D} =  D_{n+3}  + ⋯
⋮  =  ...

Choose successively polynomials p_{i}(y_{1}, ⋯, y_{i}) with the Gauss algorithm such that

E_{n,k} := A_{n} − ∑_{i=1}^{k} p_{i}A_{n+i}

and

E_{n,k}|_{D} = D_{n} + q_{1}^{k}D_{n+k+1} + q_{2}^{k}D_{n+k+2} + ⋯

with certain polynomials q_{i}^{k}.

Define E_{n} as the graph limit of the symmetric operators E_{n,k} that is, E_{n}ϕ := h if there is a sequence ϕ_{k} → ϕ, ϕ_{k} ∈ D(E_{n,k}) such that lim_{k→∞} E_{n,k}ϕ_{k} = h.

Lemma 5.3. The following statements hold:

(i) ∀f ∈ D : lim_{k→∞} E_{n,k}f = D_{n}f. (ii) D_{n} is essentially selfadjoint on D. (iii) E_{n} is equal to the selfadjoint operator D_{n}.

For f ∈ D, C ∈ W′ we get:

CE_{n,k}f = ∑_{i=0}^{n} Cp_{i}A_{i+n}f = ∑_{i=0}^{n} p_{i}A_{n+i}Cf = E_{n,k}Cf

since p_{i}, A_{n+i} are affiliated to W. Taking the limit on both sides for k → ∞ we observe:

CE_{n}f = E_{n}Cf, f ∈ D.

We take the closure and obtain because of Lemma 5.3(ii), (iii)

CD_{n} ⊆ D_{n}C, C ∈ W′,

i.e., D_{n} is affiliated to W.

The affiliated operators y_{n}, iD_{n} form an irreducible representation of the infinite Heisenberg algebra (see [S]). Therefore we have W′ = {λI}. This completes the proof of the theorem.

References

[K] Köthe, G., ,,Topologische lineare Räume“, Band 1, (Grundlehren Math. Wiss. Bd. 107), Springer, Berlin Heidelberg New York, 1960.

[K1] Kosyak, A.V., Irreducibility criterion for regular Gaussian representations of groups of infinite upper triangular matrices, Functional Anal. Appl. 24:3 (1990), 243–245.

[K2] Kosyak, A.V., Criteria for irreducibility and equivalence of regular Gaussian representations of groups of finite upper-triangular matrices of infinite order, Selecta Math. Sov. 11:3 (1992) 241–291.

[Os] Ostrovskij, V.L., Construction of quasiinvariant measures on a class of non locally compact groups, (Russian), Ukr. Mat. J. 38:2 (1986) 255–257.

[R-S] Reed, M., and Simon, B., “Methods of modern mathematical physics”, Academic Press, New York London, 1972.

[S] Samoilenko, J.S., “Spectral theory of families of selfadjoint operators”, (Russian), Naukovo Dumka, Kiev, 1984.

[S-F] Shilov, G.E., and Fan Dyk Tin, “Integral, measure and derivative on linear spaces”, (Russian), Nauka, Moscow, 1967.

[W] Warner, G., “Harmonic analysis on semi-simple Lie groups”, vol. 1,2 (Grundlehren Math. Wiss. Bd. 188, 189), Springer, Berlin Heidelberg New York, 1972.

Fachbereich Mathematik/Informatik Universität Leipzig Augustusplatz 10 D-04109 Leipzig, Germany leitenberger@mathematik.uni-leipzig.d400.de

Received August 17, 1993