Commuting Differential Operators in Positive Characteristic

Abstract

The centralizer of a noncentral element of the first Weyl algebra over a field is commutative. We discuss several proofs of this result, including a recent proof for positive characteristic fields. We also construct a large class of maximal commutative subalgebras, of the algebra differential operators on the affine line in positive characteristic. The later chapters are dedicated to the special linear group and its distribution algebra in positive characteristic. Several conjugation invariant distributions are demonstrated and their nilpotency is determined. Lastly we define a new product on the coordinate algebras of the Frobenius kernels, inspired by finding conjugation invariants.