Centralizers of differential operators of rank h

Abstract

In the paper “Burchnall-Chaundy bundles” ((1998) [12]) the second author conjectured that the centralizer of a differential operator L=x−nδ(δ−m)(δ−2m)...(δ−m(n−1)) where δ=xddx is generated by operators L and B=x−mδ(δ−n)(δ−2n)...(δ−n(m−1)) and therefore has rank equal to the greatest common divisor h of m and n. In this note we will show that this is indeed the case if the ground field K has characteristic zero. Here we restrict ourselves to purely algebraic considerations; the reader interested in geometric aspects is advised to see the paper mentioned above.