| Research Abstract |
During this period, the following results were obtained :
Subsequently I have considered unitary representations of the group of diffeomorphisms on smooth manifolds M. I denote a group of these diffeomorphisms with compact support by Diff_o(m). The group has many informations of the original manifold M, and it has close connections with Quantum Dynamics, so it is interesting to investigate representations of Diff_o(M). In fact, previously, various authors have studied and constructed many interesting unitary representations and their linear versions, most of them were irreducible.
In my research, I have obtained a series of new representations via restricted product of smooth measures with infinite mass, which is essentially inequivalent from what have been obtained now. More exactly, let E:={E_n} be a countable family(which is called μ-unital) of Borel sets in M that has the following three properties.
(1)0<μ(E_n)<+∞(2)Σ|1-μ(E_n)|<+∞(3)E_n are mutually disjoint.
Using μ-unital E, firstly
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we have a restricted product measure ν_E on M^∞. Secondly, we take an irreducible unitary representations Π of the infinite symmetric group whose element σ permutes only a finite number of natural numbers, and consider measurable functions f on M^∞ that have the properties ;
(1)f(xσ)=Π(σ)^<-1>f(x) (2)f(x) is square summable on D_E, where D_E is a Borel set such that the sets D_Eσ obtained from the action of σ on D_E are mutually disjoint, and its union is full measure with respect to ν_E. Let us denote a set of such f by H(Σ), and induce natural representation on L^2 obtained from the diagonal action of Diff_o((M) on H(Σ), whereΣ=(E,II). Then we have a unitary representation (T(g),H(Σ)), g∈
Diff_o((M). The main results on the representation is as follows :
[1](T(g),H(Σ)) is irreducible.
[2]Two representations onΣ=(E,IT) and Σ=(E',II') are equivalent, if and only if there exists a permutation(may be an infinite one) a auch that II and II' are equivalent through a, and Σ|μ(E'_<a(n)>-μ(E_n) |<+∞.
One more main result is to show that every unitary representations of Diff_o((M) has an irreducible decomposition under a fairly mild condition. The proof consists of analyzing Mautner's result on classical locally compact groups and applying the Shavgulidze measure on the diffeomorphism groups. Less
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