Infinite Dimensional Representations and Related Topics

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In 2000 the following results are obtained : 1. Let M be a smooth compact manifold and Diff^k (M) be the group of all C^k diffeomorphisins on M. Then there exists a probability measure μ on Diff^k (M) that is quasi-invariant under the left action of the smooth diffeomorphisms.2. Every continuous unitary representation of the group of smooth diffeoinorphisros on M has a dense setof the smooth vectors under a natural hypothesis.3. The above result is naturally extended to non compact M.Subsequently during the period of 2001 we considered direct questions of the above results and related topics, for example : 1. Irreducibility of the natural representation derived from the measure μ. 2. Ergodicity of μ itself.Unfortunately we have neither definite conclusions for these questions up to now nor nice applications to current algebra appeared in the second quantization in quantum mechanics.On the other hand our considerations to inductive limit of topological groups are fairly developed in this period. We are now arguing inductive limit of topological spaces, their direct products and problems related algebraic structures. Several results are obtained till now (See [T. Hirai et al. ]).