Twisted endoscopy and reducibility of induced representations for p-adic groups

Abstract

1. Introduction. This is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the Levi factors ofmaximal parabolic subgroups of p-adic groups on the unitary axis. The problem is equivalent to determining certain local Langlands L-functions which are of arithmetic significance by themselves. One of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable. When the representation is supercuspidal, we also study the reducibility of the induced representations off the unitary axis. Let F be a p-adic field of characteristic zero. Fix a positive integer n > 1 and let G be either of the groups SP2n, S02n or S02n+l. In all three cases there is a conjugacy class of maximal parabolic subgroups which has GL,, as its Levi factor. Let P MN be the standard parabolic subgroup ofG in this conjugacy class. Then M GL,,. Let tr be a discrete series representation of M GLn(F) and, given s C, let I(s, tr) be the representation of G G(F) induced from tr (R) Idet( )ls. Let I(tr) I(0, tr).