A canonical section method for conjugacy classes of GL_n(O_2)

Abstract

Let O be the ring of integers of a non-Archimedean local field with finite residue field, let p be its maximal ideal, and let O_2 = O/p^2. We show that the class equation of GL_n(O_2) depends on O only through the cardinality of its residue field: for two such rings O and O&#39; with isomorphic finite residue fields, there is a canonical bijection between the conjugacy classes of GL_n(O_2) and of GL_n(O&#39;_2) which preserves the size of every class. The argument constructs a section of the reduction map GL_n(O_2) -> GL_n(O_1) which is multiplicative on the centralizer of any element in its block Jordan canonical form, and uses it to transport the classification of conjugacy classes lying above a fixed class of GL_n(O_1) from one ring to the other. This note records the original argument for this result, developed in the author&#39;s 2010 doctoral thesis, which predates and is independent of two later proofs of closely related statements: the Ext-theoretic classification of similarity classes for n <= 4 by Prasad, Singla and Spallone, and the Hom-theoretic approach of Jambor and Plesken for general uniserial rings of length two. We record the centralizer-section argument here since it seems to be of independent interest and several colleagues have asked to see it in print.