Extension of the Virasoro and Neveu-Schwarz algebras and generalized Sturm-Liouville operators (Q1357223)

The authors consider matrix Sturm-Liouville operators of the form \[ L= \left [\begin{matrix} aD^2 +n(x), & bD+ v(x)\\ -bD +v(x), & c\end{matrix} \right] \text{ where } D= {d \over dx} \] and \(a,b,c\) are constants. Then they define an action of the Lie algebra \(\text{Vect} (S^1) \ltimes C^\infty (S^1)\) on the space of those operators. They prove that this explicitly defined action coincides with the coadjoint action of the Lie algebra \({\mathfrak g}\) that is the universal central extension of \(\text{Vect} (S^1) \ltimes C^\infty (S^1)\). The results are somewhat similar to those established before by A. A. Kirillov and G. B. Sigal dealing with the connection of Sturm-Liouville operators and the coadjoint action of the Virasoro algebra.