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On distinguished representations. (English) Zbl 0725.11026
Let $$E/F$$ be a (separable) quadratic extension of global fields with char $$F\neq 2$$, and $${\mathbb{A}}_ X,{\mathbb{A}}_ X^{\times}$$ the X-adèles, X- idèles. Put $$\underline G=GL(2)$$. A cuspidal $$\underline G({\mathbb{A}}_ E)$$-module $$\pi'$$ is called distinguished if there is a form $$\phi \in \pi'$$ with $$\int_{\underline G(F)\underline Z({\mathbb{A}}_ F)\setminus \underline G({\mathbb{A}}_ F)}\phi(h)dh\neq 0$$. These $$\pi'$$ are characterized in the author’s paper “Twisted tensors and Euler products” [Bull. Soc. Math. Fr. 116, 295-313 (1988; Zbl 0674.10026)], by a property of their “twisted-tensor” L-function $$L(s,\pi',r)$$, that it has a pole at $$s=1$$. In this paper the distinguished $$\pi'$$ are characterized as the image of the unstable base-change lifting from the unitary group $$U=\underline U(F)=U(2,E/F)$$ of $$g\in \underline G(E)$$ with $$\sigma (g)=g$$. We put $$\sigma (g)=w^ t\bar g^{-1}w^{-1}$$, $$w=$$ antidiag$$(1,-1)$$, $$t=$$ transpose, bar $$=$$ Galois action on entries of g. The underlying unstable base-change homomorphism $$b_{\kappa}$$ goes from the dual group $$\hat U=\underline G({\mathbb{C}})\rtimes W_{E/F}$$ of U to that $$\hat G'=(\underline G({\mathbb{C}})\times \underline G({\mathbb{C}}))\rtimes W_{E/F}$$ of $$G'=\underline G(E)$$. The Weil group $$W_{E/F}$$ acts via its quotient $$Gal(E/F)$$, whose non-trivial element $$\sigma$$ acts on the connected component $$\hat G^{'0}$$ by $$\sigma(g_ 1,g_ 2)=(\sigma g_ 2,\sigma g_ 1)$$, and by $$\sigma g=w^ tg^{-1}w^{-1}$$ on $$\hat U^ 0=\underline G({\mathbb{C}})$$. The map $$b_{\kappa}$$ is defind using a character $$\kappa$$ of $${\mathbb{A}}^{\times}_ E/E^{\times}$$ whose restriction to $${\mathbb{A}}_ F^{\times}$$ has the kernel $$F^{\times}N_{E/F}{\mathbb{A}}^{\times}_ E$$, as follows: $$b_{\kappa}(g)= (g,\sigma g)$$ $$(g\in \underline G({\mathbb{C}}))$$; $$b_{\kappa}(z)= (\kappa(z),\kappa(\bar z))z$$ $$(z\in W_{E/E}={\mathbb{A}}^{\times}_ E/E^{\times})$$; $$b_{\kappa}(\sigma)=$$ $$(I,-I)\sigma.$$ Here I $$=$$ identity in $$\underline G({\mathbb{C}})$$. Let $$\omega$$ be a unitary character of $${\mathbb{A}}^{\times}_ E/E^{\times}N_{E/F}{\mathbb{A}}^{\times}_ E$$ which is non-trivial on $${\mathbb{A}}_ F^{\times}$$. Then $$\omega'(z)=\omega (z/\bar z)$$ is a unitary character of $${\mathbb{A}}^{\times}_ E/E^{\times}{\mathbb{A}}_ F^{\times}$$. Put $$\kappa'(z)=\kappa(z/\bar z)$$. The main result, Theorem 1, asserts: (1) Every distinguished cuspidal $$\underline G({\mathbb{A}}_ E)$$-module $$\pi'$$ with central character $$\omega'{\kappa'}^ 2$$ is the unstable base-change lift (via $$b_{\kappa})$$ of a cuspidal non-degenerate $$\underline U({\mathbb{A}}_ F)$$-module $$\pi$$ with central character $$\omega$$. (2) Every cuspidal non- degenerate $$\underline U({\mathbb{A}}_ F)$$-module $$\pi$$ with central character $$\omega$$ lifts via the unstable map $$b_{\kappa}$$ either to a distinguished cuspidal $$\underline G({\mathbb{A}}_ E)$$-module $$\pi'$$ with central character $$\omega'{\kappa'}^ 2$$, or to an induced $$\underline G({\mathbb{A}}_ E)$$-module $$I(\mu_ 1,\mu_ 2)$$, where $$\mu_ i$$ are unitary characters of $${\mathbb{A}}^{\times}_ E/{\mathbb{A}}_ F^{\times}E^{\times}$$ with $$\mu_ 1\neq \mu_ 2$$ and $$\mu_ 1\mu_ 2=\omega'{\kappa'}^ 2.$$
In the second half of the work, from Theorem 7 on, a local analogue of this global result, also in the context of $$GL(2)$$, is studied. The introduction formulates and motivates a conjectural generalization to the context of GL(n) (including the case of $$GL(1))$$, which the author believes might be provable using the techniques employed here in the preliminary case of GL(2). Indeed this conjecture is reduced in the author’s preprint “Distinguished representations and a Fourier trace formula”, Bull. Soc. Math. Fr. (1992), to a local technical conjecture.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 22E50 Representations of Lie and linear algebraic groups over local fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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