Rings of generalized energy representations (Q1112192)

It is proved that to each Riemannian flag F of a smooth manifold X with finite dimension there is associated not only the basic energy representation \(U_ F\) [\textit{J. Marion}, \textit{D. Testard}, J. Funct. Anal. 76, No.1, 160-175 (1988; Zbl 0652.53033)] but also a set R(F) of nonlocated and order 1 unitary representations of the current group \(C_ 0^{\infty}(X,G)\), G being a compact semisimple Lie group. R(F) is a ring of representations, which may be parametrized by the union of the unitary duals of all the symmetric groups \(S_ n\) (n positive integers). \(U_ F\in R(F)\) corresponds to the unit representation of \(S_ 1\).