Filtrations of smooth principal series and Iwasawa modules

Abstract

Let $G$ be a reductive $p$-adic group‎. ‎We consider the general question‎ ‎of whether the reducibility of an induced representation can be detected in a‎ ‎``co-rank one‎" ‎situation‎. ‎For smooth complex representations induced from supercuspidal‎ ‎representations‎, ‎we show that a sufficient condition is the existence of a subquotient‎ ‎that does not appear as a subrepresentation‎. ‎An important example is the Langlands' quotient‎. ‎In addition‎, ‎we study the same general question for continuous principal series on $p$-adic Banach spaces‎. ‎Although we do not give an answer in this case‎, ‎we describe‎ ‎a related filtration on the corresponding Iwasawa modules‎.<br />