Vol. 110, No. 2, 1984

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Endoscopic groups and base change CR

Diana Shelstad

Vol. 110 (1984), No. 2, 397–416

We consider a real reductive group G with complex points G(C), Galois automorphism σ, and real points G(R) = {g G(C) : σ(g) = g}. In general, an irreducible admissible representation Π of G(C) equivalent to its Galois conjugate Π σ need not be a lift from G(R), even if G is quasi-split over R. Following the results of L-indistinguishability we might expect this phenomenon to be related to the fact that σ-twisted conjugacy on G(C) need not be “stable”, and therefore attempt to match the various “unstable” combinations of σ-twisted orbital integrals on G(C) with stable orbital integrals on certain groups H(R). The principle of functoriality in the L-group would then suggest, with reservations in the nontempered case, a relation between the σ-twisted characters of representations of G(C) fixed up to equivalence by σ and the “dual lifts” to G(C) of stable characters on the groups H(R).

In this paper we define the relevant groups H they turn out to be the endoscopic groups from L-indistinguishability... and prove a matching theorem for orbital integrals. As a preliminary to the proposed dual liftings of characters we also study the “factoring” of Galois-invariant Langlands parameters for G(C).

Mathematical Subject Classification 2000
Primary: 22E45
Secondary: 22E46
Received: 20 April 1982
Published: 1 February 1984
Diana Shelstad