Representation theory and measure theory of infinitedimensional groups and related topics
Project/Area Number  16540162 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Kochi University 
Principal Investigator 
SHIMOMURA Hiroaki Kochi University, Faculty of Education, Professor, 教育学部, 教授 (20092827)

Project Period (FY) 
2004 – 2005

Project Status 
Completed(Fiscal Year 2005)

Budget Amount *help 
¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 2005 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 2004 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  manifold / diffeomorphism group / unitary representation / infinite permutation group / positivedefinite function / irreducible decomposition / extreme decomposition / 滑らかな多様体 / diffeomorplism / 測度の制限直積 / 既約表現 / 既約分解可能性 
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 quasiinvariant 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 positivedefinite functions, I observed the difference between irreducible decomposition and extreme one. As is wellknown, 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

Report
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Research Products
(6results)