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Injectivity properties of liftings associated to Weil representations. (English) Zbl 0624.22012
As in his previous paper [ibid. 51, 333-399 (1984; see the preceeding review)], the author studies properties of the Weil lifting between the dual reductive pair (Sp, O(Q)), which in this case is given in concrete terms by a kernel function $$\theta_{\phi}(G,g)$$ on $$Sp_ n\times O(Q)$$ associated to the Weil representation and to a Schwartz function $$\phi$$ on the space of $$m\times n$$ matrices, m the dimension of the nondegenerate quadratic form.
Let $$L^ 2_{cusp}(O(Q))$$ [resp. $$L^ 2_{cusp}(Sp_ n)]$$ denote the adelic space of cusp forms on the orthogonal group O(Q) [resp. the symplectic group $$Sp_ n]$$. Moreover, let R(Q) [resp. I(Q)] be the orthogonal part in $$L^ 2_{cusp}(O(Q))$$ [resp. $$L^ 2_{cusp}(Sp_ n)]$$ to the subspace of cusp forms whose lift in $$L^ 2(Sp_ 1)$$ [resp. $$L^ 2(O(Q))]$$ vanishes. The paper presents, in the case $$n=1$$ and Q anisotropic of dimension $$>6$$, a complete description of the representations occuring in I(Q) and shows that the lift of R(Q) in $$L^ 2_{cusp}(Sp_ 1)$$ has as image the space I(-Q).
The method is to get a relation between the inner product of two lifted forms $$\ell_{\phi,\psi}$$ on O(Q) and the initial inner product of the $$\psi$$ on $$Sp_ n$$, namely: $$<\ell_{\phi,\psi_ 1}| \ell_{\phi ',\psi_ 2}>=<\psi_ 1*{\mathcal L}(\phi,\phi ')| \psi_ 2^{\Lambda}>$$, where $${\mathcal L}(\phi,\phi ')$$ is an $$L^ 1$$ function of $$Sp_ n$$ and $$\Lambda$$ a certain involution on $$L^ 2_{cusp}(Sp_ n)$$. Then the term $$<\psi *{\mathcal L}(\phi,\phi ')| \phi^{\Lambda}>$$ is shown to be the product $$L_ 1\cdot L_ 2$$, where $$L_ 1$$ is the special value of a Langlands L-function associated to the representation $$\pi$$ which carries $$\psi$$, and $$L_ 2$$ is a finite product of local integrals over the primes v where the local component $$\pi_ v$$ is not a spherical representation. In the case $$n=1$$, the author establishes a formula which shows that $$L_ 1\neq 0$$; the nonvanishing of $$L_ 2$$ depends on whether the components $$\pi_ v$$ occur in the associated local Weil representations. Then using results of a paper by the author and G. Schiffmann [Mem. Am. Math. Soc. 231 (1980; Zbl 0442.22006)], the description of I(Q) is obtained.
On the other hand, the use of the inner product formula gives a comparison of traces of Hecke operators acting, respectively, on the space of modular cusp forms of weight 4t and on the image in $$L^ 2_{cusp}(O(Q))$$ of the lifting. It also provides a way to determine the algebraic nature of the ratio $$<\ell_{\phi,\psi_ 1}| \ell_{\phi ',\psi_ 2}>/<\psi_ 1| \psi_ 2^{\Lambda}>$$ when $$\psi_ 1$$ is an eigenfunction of $${\mathcal L}(\phi,\phi ').$$
Finally, as a third application, the author gives a construction of a global cuspidal automorphic representation of O(Q) such that a local component at infinity has nonvanishing Lie algebra cohomology of a certain described level.

##### MSC:
 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11F27 Theta series; Weil representation; theta correspondences 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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