2005 Fiscal Year Final Research Report Summary
Representation theory and measure theory of infinite-dimensional groups and related topics
| Project/Area Number |
16540162
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kochi University |
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Principal Investigator |
SHIMOMURA Hiroaki Kochi University, Faculty of Education, Professor, 教育学部, 教授 (20092827)
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| Project Period (FY) |
2004 – 2005
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| Keywords | manifold / diffeomorphism group / unitary representation / infinite permutation group / positive-definite function / irreducible decomposition / extreme decomposition |
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| Research Abstract |
In the first half period, I considered the representations of the group of diffeomorphisms denoted by Diff_0(M) on smooth manifolds M.Historically we have many analysis on this group, however my researching object is a natural representation on L^2 space overM^∞ derived from a restricted product measure ν_E of a smooth measure on M with infinite mass. ν_E is quasi-invariant under the diagonal action of Diff_0(M), and hence we have a natural representation T of Diff_0(M) over the L^2 space.
Secondly, take a unitary representation II of the infinite permutation group S of the finite permutations of the natural numbers, and take functions f on M^∞ that have properties (1) f(xσ)=II(σ)^<-1>f(x) (2) f(x) is square summable. Let H(Σ) be the space of all such f.
We introduce another natural representation of Diff_0(M) on this space, similarly as above. Put Σ:=(E,II). Then we have unitary representations (T(g), H(Σ)), g∈ Diff_0(M), and we have already known that these representations are all irre
… More
ducible.
In the present research, I investigated the irreducible components of the representation T, and clarified that they are nothing but the above (T(g), H(Σ)).
In the later half period, I turned my attention to the applications of Diff_0(M). For example, it is interesting to find functional equations through the representations of Diff_0(M) via analysis of the infinite permutation group or the theory of asymptotic behavior of Young diagrams. Another one is an approach to a realization of irreducible component of the regular representation of S through the representations of Diff_0(M).
After reading preceding bibliographies, in particular that by E Thoma, and further considering extreme decomposition of positive-definite functions, I observed the difference between irreducible decomposition and extreme one.
As is well-known, extreme decomposition contains irreducible ones of the corresponding unitary representation, and besides another decomposition like the Choqet theorem. In this period, I studied how these two concepts connect with each other and obtained a result for the time being. However it is inconvenient for applications of this result to concrete problems, so I am trying to improve it more useful forms. I hope it will be useful in the analysis of the regular representation of S. Less
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