Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

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Journal of Functional Analysis 236 (2006) 634–681 www.elsevier.com/locate/jfa Quasiregular representations of the infinite-dimensional nilpotent group Sergio Albeverio a,1 , Alexandre Kosyak b,∗ a Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany b Institute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka, Kyiv 01601, Ukraine Received 7 February 2006; accepted 13 March 2006 Communicated by Paul Malliavin Abstract In the present work an analog of the quasiregular representation which is well known for locally-compact groups is constructed for the nilpotent infinite-dimensional group BN 0 and a criterion for its irreducibility is presented. This construction uses the infinite tensor product of arbitrary Gaussian measures in the spaces Rm with m>1 extending in a rather subtle way previous work of the second author for the infinite tensor product of one-dimensional Gaussian measures. © 2006 Elsevier Inc. All rights reserved. Keywords: Infinite-dimensional groups; Nilpotent groups; Quasiregular representations; Irreducibility; Infinite tensor products; Gaussian measures; Ismagilov conjecture 1. Introduction 1.1. The setting and the main results Let (X, B) be a measurable space and let Aut(X) denote the group of all measurable automorphisms of the space X. With any measurable action α : G → Aut(X) of a group * Corresponding author. Fax: 38044 2352010. E-mail addresses: albeverio@uni.bonn.de (S. Albeverio), kosyak01@yahoo.com, kosyak@imath.kiev.ua (A. Kosyak). 1 SFB 611, BIGS, IZKS, Bonn, BiBoS, Bielefeld–Bonn, Germany; CERFIM, Locarno; Accademia di Architettura, USI, Mendrisio, Switzerland. 0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2006.03.013
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 635 G on the space X and a G-quasi-invariant measure μ on X one can associate a unitary representation π α,μ,X : G → U(L2 (X, μ)), of the group G by the formula (π α,μ,X t f )(x) = (dμ(αt−1(x))/dμ(x)) 1/2f(αt−1(x)), f ∈ L2 (X, μ). Let us set α(G) ={αt ∈ Aut(X) | t ∈ G}. Let α(G) ′ be the centralizer of the subgroup α(G) in Aut(X): α(G) ′ ={g ∈ Aut(X) |{g,αt}= gαtg−1α −1 t = e ∀t ∈ G}. The following conjecture has been discussed in [23–25]. Conjecture 1. The representation π α,μ,X : G → U(L 2 (X, μ)) is irreducible if and only if : (1) μ g ⊥ μ ∀g ∈ α(G) ′ \{e} (where ⊥ stands for singular), (2) the measure μ is G-ergodic. We recall that a measure μ is G-ergodic if f(αt(x)) = f(x)∀t ∈ G implies f(x)= const μ a.e. for all functions f ∈ L1 (X, μ). In this paper we shall prove Conjecture 1 in the case where G is the infinite-dimensional nilpotent group G = BN 0 of finite upper-triangular matrices of infinite order with unities on the diagonal, the space X = Xm being the set of left cosets Gm \ BN , Gm being suitable subgroups of the group BN of all upper-triangular matrices of infinite order with unities on the diagonal, and μ an infinite tensor product of Gaussian measures on the spaces Rm with some fixed m>1. A more detailed explanation of the concepts used here is given in the following sections. 1.2. Regular and quasiregular representations of locally compact groups Let G be a locally compact group. The right ρ (respectively left λ) regular representation of the group G is a particular case of the representation π α,μ,X with the space X = G, the action α being the right action α = R (respectively the left action α = L), and the measure μ being the right invariant Haar measure on the group G (see, for example, [8,16,17,37]). A quasiregular representation of a locally compact group G is also a particular case of the representation π α,μ,X (see, for example, [37, p. 27]) with the space X = H \ G, where H is a subgroup of the group G, the action α being the right action of the group G on the space X and the measure μ being some quasi-invariant measure on the space X (this measure is unique up to a scalar multiple). We remark that in [16,17] this representation has also been called geometric representation. 1.3. Analogs of the regular and quasiregular representations of infinite-dimensional groups and the Ismagilov conjecture In the present article we will consider the approach which deals with analogs for infinitedimensional groups of the regular and quasiregular representations of finite-dimensional groups. Let G be an infinite-dimensional topological group. To define an analog of the regular representation, let us consider some topological group G, containing the initial group G as a dense subgroup, i.e. G = G (G being the closure of G). Suppose we have some quasi-invariant measure μ on X = G with respect to the right action of the group G, i.e.α = R, Rt(x) = xt −1 .Inthis case we shall call the representation π α,μ,G an analog of the regular representation. We shall denote this representation by T R,μ , and the Conjecture 1 is reduced to the following Ismagilov conjecture.
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