Analytic and geometric studies of group representations and their applications

点击翻译按钮获取中文摘要

The Head Investigator Nomura, studying algebraic systems by functional-analytic method, has investigated the Riemann-Hilbert manifold of idempotents (Grassmann manifolds) of Hilbert Jordan algebras and described geodesics, the Riemannian metric and the sectional curvature formula by using the Jordan-algebra structure, clarifying the contribution of the algebra. Spectral decomposition of Berezin transformation associated to the multiplicity-free compact Lie group actions has also been studied. Some of the results are already reported at the work-shop in Tottori and in E.Fujita's master thesis (Kyoto university).Investigator Umeda has studied the invariant theory and the representation theory. A particular focus has been placed upon the research on the the Capelli identity associated to dual pairs under the theme "invariant theory based on quantum symmetry". A formula for the Capelli identity associated to the dual pair $O_-n, {/goth sp} _- {Zm} $ has been obtained. This explains some of the Capelli identities associated to multiplicity-free actions and also the central elements of the universal enveloping algebra of $ {/goth o} _-n} $. Though not an ultimately complete form, the results are reported in G.Ochial's master thesis (Kyoto University).Investigator Hiral has treated the infinite symmetric groups and the diffeomorphism group of manifolds. Considering the infinite tensor product of natural representations, one finds that a certain infinite symmetric group acts as intertwining operators. It is shown that these two kinds of groups form a dual pair in a certain case and investigation has been done on the irreducible decomposition of this infinite tensor product through the dual pair.