Local coefficients as Artin factors for real groups

Abstract

Introduction. The purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi-split real reductive algebraic groups in [27] with their corresponding Artin factors attached by local class field theory [21]. As a consequence, we establish an identity satisfied by certain normalized intertwining operators. It seems to be useful in applications of the trace formula [1, 29]. More precisely, let G be the group of real points of a quasi-split reductive algebraic group over R. Let A be the set of simple roots defined by a fixed minimal parabolic subgroup Po = M0A0U of G. Fix 6 A, and let P = P9 be the corresponding standard parabolic subgroup of G and write P = MAN for its Langlands decomposition. Fix a nondegenerate character x °f U i ^ (a, H{o)) be an irreducible admissible x-gic Banach (in particular x~generic unitary) representation of M (cf. Section 1). Given v G a£, the complex dual of the Lie algebra of A, let I(v,o,0) be the continuously (quasi-unitarily, if o is unitary) induced representation IndP^Ga® e v 9 and let V(v,o,0) be its space (Section 0). Then F ^ a , ^ = Vfoo^O). Now, let W be the Weyl group of Ao in G. Choose w G W such that w(0) A. Let N~ be the unipotent group opposite to N. Define N$ = U C wN~w~ where w is a representative of w in G. For/ G F(^,a,^)00, define