Character identities in the twisted endoscopy of real by Paul Mezo

By Paul Mezo

Think G is a true reductive algebraic workforce, ? is an automorphism of G, and ? is a quasicharacter of the gang of genuine issues G(R). lower than a few extra assumptions, the idea of twisted endoscopy affiliates to this triple genuine reductive teams H. The neighborhood Langlands Correspondence walls the admissible representations of H(R) and G(R) into L-packets. the writer proves twisted personality identities among L-packets of H(R) and G(R) constructed from crucial discrete sequence or limits of discrete sequence

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9 [Kna96]). This non-uniqueness may mean that additional roots of unity are possible in the normalization of U. However, the possibility of several extensions to L will be accounted for in our character identities and we have chosen not to include the roots of unity in the notations for ¯ 1 , U1 and U. 1 such that τ is equivalent to ¯ 1 . 1 by setting L0 = Gder (R)0 , l = gder , H = Sder (R), and h = sder . 1 we have fixed a maximal compact subgroup K ⊃ Sder (R) of Gder (R)0 and a positive system for R(k⊗C, sder ⊗C).

57) ϕH (w) ∈ ξH (c(w))H 1 1 1 1 This proves ϕH1 (WR ) ⊂ ξH1 (H). Lemma 10. The map from ϕH1 to ϕ∗ described above is well-defined. Proof. We are to prove that the map does not depend on the choice of representative ϕH1 in its definition. This is shown in §2 [She10]. We shall provide a more detailed argument here. Suppose then that ϕH1 ∈ ϕH1 also satisfies p ◦ ϕH1 = p ◦ ξH1 ◦ c. 33 34 6. SPECTRAL TRANSFER FOR SQUARE-INTEGRABLE REPRESENTATIONS ˆ 1 such that ϕ = Int(h) ◦ ϕH and This means that there is some h ∈ H 1 H1 p(h)p(ϕH1 (w))p(h)−1 = p(ϕH1 (w)) = p(ξH1 (c(w))) = p(ϕH1 (w)), w ∈ WR .

It follows from (40) that for w ∈ Ω(Gder (R)0 , Sder (R))δθ , x ˜ ∈ L(Λ1 ) which projects to x ∈ Sder (R) and 28 5. TEMPERED ESSENTIALLY SQUARE-INTEGRABLE REPRESENTATIONS X ∈ sder such that x = exp(X) we have τ w ρ˜1 (x2 ) = τ0w (ρ˜1 −1 )w ρ˜1 (x2 ) x2 ) = τ0w (ρ˜1 −1 )w ρ˜1 (˜ = ewiΛ1 (2X) ew2ρ1 (X) det(Adwx−1 w−1 )|u det(Adx)|u = ewiΛ1 (2X) det(Adwxw−1 )|u det(Adwx−1 w−1 )|u eρ1 (2X) = e(wiΛ1 +ρ1 )(2X) . As every element of the compact connected torus Sder (R) is a square, this proves that τ w ρ˜1 (x) = τ0w (ρ˜1 −1 )w ρ˜1 (x) = e(wiΛ1 +ρ1 )(X) .

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