By Robert P. Langlands
The matter of base swap or of lifting for automorphic representations might be brought in different methods. It emerges in a short time whilst one pursues the formal rules expounded within the article 20 that may actually be diminished to 1, viz., the functoriality of automorphic types with appreciate to what's now referred because the L-group.
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Extra resources for Base change for GL(2)
A1 (γ, fλ ) is certainly 0 unless λ = (m + r, m − r), r ≥ 0. If this condition is satisfied it equals 2 n| |(r + α)q r 1− 1 q meas G(O) , meas A(O) r > 0, or 2q −α n| | meas G(O) meas A(O) α j=0 jq j 1 q , 1− 1 q 1− The contribution 2 n| | meas G(O) r rq meas A(O) r = 0. q r meas G(O) Base change 38 is accounted for by the first of the three summands in the lemma. The contribution 2 n| | meas G(O) r αq meas A(O) 1− 1 q by the second, and the remainder, which is 0 for r > 0 and 2 n| | meas G(O) meas A(O) α−1 j=0 jq j−α 1− 1 q for r = 0, by the third.
P) ........ p0 X Since X is the set of fixed points of σ in X(E), the paths from p0 to p and from p0 to σ(p) must start off in different directions. In other words the initial edge of the path from p0 to p does not lie in X . This shows that there are q r (1 − q 1− ) possibilities for the p or, what is the same, the p occurring in the above sum if r > 0 and just 1 if r = 0. 3. Since λ(γ) = λ(δ), the integral is certainly 0 unless λ = (m + r, m − r). If this condition is satisfied it equals mE (λ) 1 + q −1 −r q 2 meas G(O) 1 + q − times 1 − z −1 1−z 1 − q− z− · · z − −1 −1 1−z 1−q z 1 − q −1 z 1 2πi r + 1 − q− z 1 − z −1 1−z · · z− −1 −1 1−z 1−q z 1 − q −1 z r dz .
We have already seen that H is isomorphic to the representation ring of GL(2, C). With G = G(E/F ) we form the direct product L G = GL(2, C) × G which is the L-group of G. Let Φ be the Frobenius element in G. The representation ring of GL(2, C) is isomorphic, by means of the map g → g × Φ from GL(2, C) to GL(2, C) × Φ ⊆LG, to the algebra H obtained by restricting to GL(2, C) × Φ the representation ring of LG, which is the algebra of functions on LG formed by linear combinations of characters of finite-dimensional complex analytic representations of LG.