Prehomogeneity on Quasi-Split Classical Groups and Poles of Intertwining Operators

http://dx.doi.org/10.4153/CJM-2009-037-6
Canad. J. Math. 61(2009), 691-707

  Published:2009-06-01
 Printed: Jun 2009

Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit, connected, reductive classical group $G$ defined over a non-Archimedean field and $A$ is the standard intertwining operator attached to a tempered representation of $G$ induced from $M$. In this paper we determine all the cases in which $\Lie(N)$ is prehomogeneous under $\Ad(m)$ when $N$ is non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$.