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The basic representation of the current group \(O(n,1)^X\) in the \(L^2\) space over the generalized Lebesgue measure. (English) Zbl 1147.22013

Let \(X\) be a smooth manifold, endowed with a finite continuous measure \(m\), and \(G\) be a Lie group. The current group \(G^X\) is the set of bounded \(G\)-valued Borel functions \(X\to G\), which is a group with respect to pointwise multiplication. In [Funct. Anal. Appl. 8,. 151–153 (1974); translation from Funkts. Anal. Prilozh. 8, No. 2, 67–69 (1974; Zbl 0299.22004)], the authors and I. M. Gel’fand constructed for the groups \(G=\text{SO}(n,1)\) and \(G=\text{SU}(n,1)\) certain irreducible unitary representations of the current group \(G^X\) which are invariant under all \(m\)-preserving transformations of \(X\), called basic representations. Moreover, they showed that, among the simple real Lie groups \(G\), such representations exist only for the groups \(\text{SO}(n,1)\) and \(\text{SU}(n,1)\).
In the paper under review, the authors undertake a detailed study of the so-called commutative model of the basis representation of \(G^X\) for the group \(\text{O}(n,1)\). Here the term “commutative model” is understood in the following sense. If \(A\) is an abelian subgroup of the topological group \(H\) and \(\pi\) is a unitary representation of \(H\) for which the restriction to \(A\) is cyclic, then one may expect a realization of the representation of \(H\) in an \(L^2\)-space of some measure living either on the character group \(\widehat A\) of \(A\) (if some form of the Bochner theorem applies to \(A\)) or on the spectrum of the commutative \(C^*\)-algebra generated by \(\pi(A)\). Such a realization is called a commutative model of the representation \(\pi\) of \(H\). In the present paper, \(H=\text{O}(n,1)^X\) and \(A=Z^X\), where \(Z\subset \text{O}(n,1)\) is the maximal unipotent subgroup, which is actually abelian and isomorphic to \(\mathbb R^{n-1}\) (in contrast to the situation for \(\text{U}(n,1)\), where \(Z\) is a \(2\)-step nilpotent Heisenberg group).
It is a central result of this paper that the basic representation of \(\text{O}(n,1)^X\) has a commutative realization with respect to \(Z^X\) on an \(L^2\)-space of a measure \(\nu\) living on a space of \(\mathbb R^{n-1}\)-valued distributions on \(X\) which is, in addition, invariant under the action of the current group \({\text{O}(n-1)^X}\). Actually, in all cases the measure \(\nu\) is concentrated on distributions which are countable sums of \(\delta\)-functions. An important point of the commutative model is that the operators describing the representation of the current group are quite explicit, which is not the case for the known Fock space realizations of the basic representation. The key observation that leads to the commutative realization of the basic representation of \(\text{O}(n,1)^X\) is that the corresponding infinitely divisible positive definite function on \(\text{O}(n,1)\) leads to an infinitely divisible measure on the character group \(\widehat Z\cong \mathbb R^{n-1}\), hence to a Lévy process with values in this group. This measure is determined explicitly from the construction of the basic representation by a limiting process from the complementary series of unitary representations of \(\text{O}(n,1)\).

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E46 Semisimple Lie groups and their representations
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps

Citations:

Zbl 0299.22004
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References:

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