On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical

Abstract

AbstractLet P = M N be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group G over a p-adic field F. Assume that there exists w0∊ G(F) that normalizes M and conjugates P to an opposite parabolic subgroup. When N has a Zariski dense Int M-orbit, F. Shahidi and X. Yu described a certain distribution D on M(F), such that, for irreducible unitary supercuspidal representations π of M(F) withis irreducible if and only if D( f )≠ 0 for some pseudocoefficient f of π. Since this irreducibility is conjecturally related to π arising via transfer from certain twisted endoscopic groups of M, it is of interest to realize D as endoscopic transfer from a simpler distribution on a twisted endoscopic group H of M. This has been done in many situations where N is abelian. Here we handle the standard examples in cases where N is nonabelian but admit a Zariski dense Int M-orbit.