\(\text{Diff }S^ 1\) as coadjoint orbit in the Virasoro space of moments
A distribution \(T\) on \(S^ 1\) is defined by \(T(f)= \sum_{n\in\mathbb{N}} {f(n)\over 2^ n}\), where \(f\) is \(2\pi\)-periodic and differentiable. For any vector field \(\xi= f(\theta) {\partial\over \partial\theta}\) on \(S^ 1\), one has \[ \langle T\otimes d\theta\mid \xi\rangle= \sum_{n\in \mathbb{N}} {f(n)\over 2^ n}. \] The author proves that the isotropy subgroup of \(T\otimes d\theta\) for the coadjoint action of \(\text{Diff}(S^ 1)\) is reduced to the identity. So, he obtains an embedding of \(\text{Diff}(S^ 1)\) as a symplectic coadjoint orbit of itself. Also, an embedding of \(\text{Diff}(S^ 1)\) in the Virasoro group with non-zero central charge is presented.
scientific article; zbMATH DE number 569292