Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions

Abstract

We investigate PBW deformations H of k[x,y]#G where G is the cyclic group of order p and k also has characteristic p; in these deformations, [x,y] takes a value in kG. These algebras are versions of symplectic reflection algebras that only exist in positive characteristic. They also happen to possess a presentation as an Ore extension over a commutative subring R, and via the derivation defining the extension, have interesting connections to certain polynomials appearing in combinatorics and related to alternating permutations (the Andr\'e polynomials). We find the center of these algebras, their Verma modules, their simple modules, and the Ext groups between simples. The Verma modules coincide with the fibers of Mod H -->> Spec R, and Mod H turns out to be the disjoint union, in the sense of Smith and Zhang, of the Verma modules. As in non-defining characteristic, there are some distinctions between the t=0 and t=1 cases. We also obtain results when G is replaced with an elementary abelian p-group E: in particular, we find the center and the simple modules when [x,y] is in k[E].