On the Poisson geometry of the Adler-Gel'fand-Dikii brackets

In this paper, we consider a symplectic leaf that goes through a singular point of the Adler-Gel'fand-Dikii Poisson bracket associated to SL(n, R). We End a finite-dimensional transverse section Q at the singular point and we prove that one can induce a Poisson structure on Q (the transverse structure) that is linearizable and equivalent to the Lie-Poisson structure on sl(n, R)*. This problem is closely related to finding normal forms for nth order scalar differential operators with periodic coefficient. We partially generalize a well-known result for Hill's operators to the higher order case.