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).
