Type: Article
Publication Date: 1985-09-01
Citations: 183
DOI: https://doi.org/10.1215/s0012-7094-85-05252-4
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 P o = M 0 A 0 U of G. Fix 6 C A, and let P = P 9 be the corresponding standard parabolic subgroup of G and write P = MAN for its Langlands decomposition.Fix a nondegenerate character x °f U an< i (a, H{o)) be an irreducible admissible x-g ener ic 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 Ind P ^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 A o in G. Choose w G W such that w(0) C 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 = f (cf.(3.1) and (3.2) of Section 3).The convergence and meromorphic continuation of A (v, a, w)f has been studied by Knapp and Stein in [12,13] (for minimal P see also Schiffmann [24]).The representation I(p,o,8) is x-g ener i c -Let K(V,O) be the canonical Whittaker functional attached to V{v,o,9)^ by relation (2.2) of Section 2. Then the local coefficient C x (v,o,0,w) is a complex number defined by K(v,o)(f) = C x (v,oJ,w)K(w(r),w(o))(A(p,o,w)(f)), V/ G Viv.o.O)^ (cf.Section 3 and [27]).