By Steven Dale Cutkosky

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**Example text**

The torus TH (R) is elliptic. Proof. This is immediate from the assumption that δ is θ-elliptic and Lemma 12. 1. A parameterization of stable data. 1) [KS99]) geometric transfer factors of the form Δ(δH , δ) are present. In our setup, this becomes Δ(γ1 , δ ), where δ ∈ G(R) runs over a set of representatives for the θ-conjugacy classes under G(R) of elements whose norm is γ1 . Every δ is of the form x−1 δθ(x) for some x ∈ G. Our ﬁrst goal is to show that the set of representatives δ is to some extent parameterized by the set (Ω(G, S)/ΩR (G, S))δθ which we deﬁne to be the set of cosets in Ω(G, S)/ΩR (G, S) whose representatives w ∈ Ω(G, S) satisfy (64) w−1 δθ w(δθ)−1 ∈ ΩR (G, S).

Then tr Gder (R)0 1 |Gder (R)0 ∩ ZG (R)| k = v∈B r=1 Gder (R)0 23 be an orthonormal basis k f (zxδj θ)χπ (z) dz π(xδj ) U dx j=1 ZG (R) 1 |Gder (R)0 ∩ ZG (R)| k f (zxδj θ)χπ (z) dz j=1 ZG (R) · π(xδj )Uπ(δr )v, π(δr )v dx k = v∈B r=1 Gder (R)0 |Gder 1 ∩ ZG (R)| (R)0 k f (zxδj θ)χπ (z) dz j=1 ZG (R) · ω(δr−1 ) π(δr−1 xδj θ(δr ))Uv, v dx k = v∈B r=1 Gder (R)0 1 |Gder (R)0 ∩ ZG (R)| k j=1 f (zδr xδj θδr−1 )χπ (z) dz ZG (R) · ω(δr−1 ) π(x) π(δj )Uv, v dx. Here we have used (24) and a change of variable.

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 .