The stable representations of GL N over finite local principal ideal rings

Abstract

Let be a discrete valuation ring with maximal ideal and with finite residue field , the field with q elements where q is a power of a prime p. For , we write for the reduction of modulo the ideal . An irreducible ordinary representation of the finite group is called stable if its restriction to the principal congruence kernel , where , consists of irreducible representations whose stabilizers modulo , where , are centralizers of certain matrices in , called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group for . Communicated by Scott Chapman