Isolatedness of the minimal representation and minimal decay of exceptional groups

Abstract

Using a new definition of rank for representations of semisimple groups sharp results are proved for the decay of matrix coefficients of unitary representations of two types of non-split $p$-adic simple algebraic groups of exceptional type. These sharp bounds are achieved by minimal representations. It is also shown that in one of the cases considered, the minimal representation is isolated in the unitary dual.