On the structure of principal series representations of SL2(𝔽̄p)

Abstract

Let [Formula: see text] be a prime number and [Formula: see text] be the algebraic closure of the finite field [Formula: see text] with [Formula: see text] elements. Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the standard Borel subgroup of [Formula: see text]. For any character [Formula: see text] of [Formula: see text], define [Formula: see text], where [Formula: see text] is the group algebra of [Formula: see text]. In this paper, we prove two interesting properties of [Formula: see text]: (a) [Formula: see text] is either of finite length or residually finite; (b) Any nontrivial quotient of [Formula: see text] is finite dimensional.